Method and system for analytic network process (ANP) rank influence analysis

ABSTRACT

An apparatus includes an analytic network process (ANP) storage memory that stores an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model, and a processor in communication with the ANP storage memory. The process performs a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation-in-part of the following U.S. patent application Ser. Nos. 12/508,703 filed Jul. 24, 2009; 12/646,289 filed Dec. 23, 2009; 12/646,418 filed Dec. 23, 2009; 12/646,099, filed Dec. 23, 2009; 12/646,312 filed Dec. 23, 2009; and claims the benefit of the following U.S. provisional patent applications: 61/412,996 filed Nov. 12, 2010; and 61/413,182 filed Nov. 12, 2010; all of which are expressly incorporated herein by reference.

TECHNICAL FIELD

The present invention relates in general to analysis of factors in a decision, and more specifically to analysis of rank influence in an analytic network process (ANP) model.

BACKGROUND

The factors in a decision can be represented and quantified in an analytic hierarchy process (AHP) model. The AHP model can be used to relate the factors to overall goals, and to evaluate alternative solutions. Measuring sensitivity of factors in AHP trees is generally known. As a result of the measurements of sensitivity of nodes in the tree context, a user can see how influential individual nodes are, such as how sensitive the decision model's alternative scores are to changes in weight of various nodes in the AHP tree. Measuring sensitivity of the factors in a decision can be straightforward in the AHP tree because the AHP model uses a tree structure.

The factors in a decision also can be represented and quantified in an analytic network process (ANP) model. A process of decision making using an ANP model serves to break down large decisions into smaller, manageable decisions. When a decision is represented as a typical ANP model, nodes in the ANP model can be connected to each other without regard for hierarchy level so as to represent the interrelationship between the smaller decisions. The connections that represent the effect of smaller decisions can be synthesized to arrive at the ultimate decision. Measuring sensitivity of a factor in a decision quantified in the ANP model is consequently difficult since the ANP is not a simple tree structure and a change in one factor affects interrelated decisions and may (or may not) affect the ultimate decision. When using tree techniques on an ANP model, there is sensitivity. However, a different tool needs to be used to determine the sensitivity.

SUMMARY

Accordingly, one or more embodiments provide an apparatus, method, and/or computer-readable medium for analyzing ANP rank influence.

An analytic network process (ANP) storage memory stores an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model. A processor is in communication with the ANP storage memory. The processor is configured to perform a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model.

According to another embodiment, the apparatus, method and/or computer-readable medium determines a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model.

Still another embodiment indicates the nodes determined to have the most influence.

According to still another embodiment, a user is prompted to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model.

Yet another embodiment provides, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model.

Another embodiment bases the rank of each node in the ANP model on an amount of influence each said node has on rankings of alternatives in the ANP model.

In yet a further embodiment, the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model.

Another embodiment provides an input unit configured to input, from an input device to the processor, pairwise comparisons, ANP ratings, or ANP client data, which are stored as the data in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values.

Also, another embodiment is an apparatus, method and/or computer-readable medium that provides for an analytic network process (ANP) storage memory that stores an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model. The embodiment can:

perform a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model;

determine a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model;

indicate the nodes determined to have the most influence;

prompt a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model;

provide, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model;

base the rank of each node in the ANP model on an amount of influence each said node has on rankings of alternatives in the ANP model; and

and

provide an input unit configured to input, from an input device to the processor, pairwise comparisons, ANP ratings, or ANP client data, which are stored as the data in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values,

wherein the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model,

wherein the amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model is obtained by an upper rank influence score and a lower rank influence score.

Still another embodiment performs row perturbations in the weighted supermatrix representing the ANP model, so as to determine for the first p⁺ above the fixed point p₀ where an upper rank change occurs and p⁻ below p₀ where a lower rank change occurs, and using the first p⁺ and p⁻ to construct a value that indicates the upper and lower rank influence of the rows throughout the weighted supermatrix.

In yet another embodiment, the upper rank influence score rki⁺ _(A,W,r) is defined by

${rki}_{A,W,r}^{+} = \frac{1 - p_{A,W,r}^{+}}{1 - p_{0}}$ and the lower rank influence score rki⁻ _(A,W,r) is defined by

${{rki}_{A,W,r}^{-} = \frac{1 - p_{A,W,r}^{-}}{1 - p_{0}}},$

where

A is the ANP model,

W is the weighted supermatrix of A(of dimensions n×n),

1≦r≦n is a row of W,

k is column of W,

i is an alternative in A,

W(p) is a family of row perturbations for row r of W with p₀ as the fixed point,

p₀ represents returning the node values to the original weights,

p⁺ is above p₀ where the upper rank change occurs, and

p⁻ is below p₀ where the lower rank change occurs.

Another embodiment can be a method of performing any or all of the above.

Still another embodiment can be a computer readable storage medium comprising instructions for the described method.

Further, the purpose of the foregoing abstract is to enable the U.S. Patent and Trademark Office and the public generally, and especially the scientists, engineers and practitioners in the art who are not familiar with patent or legal terms or phraseology, to determine quickly from a cursory inspection the nature and essence of the technical disclosure of the application. The abstract is neither intended to define the invention of the application, which is measured by the claims, nor is it intended to be limiting as to the scope of the invention in any way.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying figures, where like reference numerals refer to identical or functionally similar elements and which together with the detailed description below are incorporated in and form part of the specification, serve to further illustrate various exemplary embodiments and to explain various principles and advantages in accordance with the present invention.

FIG. 1 is a block diagram illustrating a system for ANP rank influence analysis;

FIG. 2 is a diagram illustrating a simplified representation of an ANP model;

FIG. 3 is a block diagram illustrating portions of an exemplary computer;

FIG. 4 is an example implementation for illustrating rank influence analysis;

FIG. 5 is a flow chart illustrating a procedure to analyze rank influence of an ANP model;

FIG. 6 is a diagram illustrating a measurement of sensitivity of a node in an ANP weighted supermatrix;

FIG. 7 is an explanatory diagram for a further explanation of FIG. 6; and

FIG. 8 is a data flow diagram illustrating a measurement of change distance of nodes in an ANP weighted supermatrix.

DETAILED DESCRIPTION

In overview, the present disclosure concerns computers, computer networks and computer systems, such as an intranet, local area network, distributed network, or the like having a capability of analyzing properties of decision models. Such computer networks and computer systems may further provide services such as interacting with users, and/or evaluating modifications to a decision model. More particularly, various inventive concepts and principles are embodied in systems, devices, and methods therein related to analyzing properties of an analytic network process model. It should be noted that the term device may be used interchangeably herein with computer, wireless communication unit, or the like. Examples of such devices include personal computers, general purpose computers, personal digital assistants, cellular handsets, and equivalents thereof.

The instant disclosure is provided to further explain in an enabling fashion the best modes of performing one or more embodiments of the present invention. The disclosure is further offered to enhance an understanding and appreciation for the inventive principles and advantages thereof, rather than to limit in any manner the invention. The invention is defined solely by the appended claims including any amendments made during the pendency of this application and all equivalents of those claims as issued.

It is further understood that the use of relational terms such as first and second, and the like, if any, are used solely to distinguish one from another entity, item, or action without necessarily requiring or implying any actual such relationship or order between such entities, items or actions. It is noted that some embodiments may include a plurality of processes or steps, which can be performed in any order, unless expressly and necessarily limited to a particular order; i.e., processes or steps that are not so limited may be performed in any order.

Much of the inventive functionality and many of the inventive principles when implemented, are best supported with or in software or integrated circuits (ICs), such as a digital signal processor and software therefore, and/or application specific ICs. It is expected that one of ordinary skill, notwithstanding possibly significant effort and many design choices motivated by, for example, available time, current technology, and economic considerations, when guided by the concepts and principles disclosed herein will be readily capable of generating such software instructions or ICs with minimal experimentation. Therefore, in the interest of brevity and minimization of any risk of obscuring the principles and concepts according to the present invention, further discussion of such software and ICs, if any, will be limited to the essentials with respect to the principles and concepts used by the exemplary embodiments.

Part I: Influence Analysis

A fundamental question for ANP models is which nodes have the most influence on the alternatives. In particular which nodes have the most influence on the rankings of the alternatives. Using ANP row sensitivity (defined in Part II) we develop a method for ranking the nodes of an ANP model in order of importance to the rankings of the alternatives.

A node in an ANP model is very influential if it changes slightly and changes the rank, and vice versa. The question is, where is the point where I get a rank change. If we identify the node where a small change affects the rank, we can then be more detailed in providing data for that node and more cursory in providing data for other nodes.

This question focuses in on ANP rank influence, changing it up and down and seeing how much that really changes the model. Picture a value between 0 and 1 (minimum and maximum values respectively), and changing the value of one node. As I change the importance of, e.g., social benefits, down to 0 or up to 1, when does a rank switch occur, for example, when does rank #1 go down to #2. If something is very risky, and as the importance of risks is pushed down, the risk penalty does not affect the ranking that much. There will be a point at which the rankings just start to switch. The system and method discussed herein provide a way to figure out where to set the value to just to get the switch, and then calculate the score based on that. A node is very influential if it has a rank change with a small change; a node which is unimportant to rank influence can be changed to 0 or 1 and cause no switch in rank and hence is not important to rank influence. Rank influence doesn't care about numbers overall, but reflects the point at which there is a rank change.

Consider an ANP model with 20 nodes in it. It is useful to determine which nodes have the most influence on the model, in terms of which nodes most affect the outcome. For example, in a standard BOCR model, what influences the alternatives scores more: benefits, opportunities, costs or risks? Which of those 4 nodes influences the ranking of the alternatives the most? Before this system and method, there was no way of figuring out the most and least influential nodes relative to the ranking A user simply had no idea how the feedback worked. This system/method allows a user to see which nodes are the most influential.

Considering, e.g., choosing schools and cost is not important so that a user just cares about ranking The question is, which node causes that ranking to change? What are the factors that really drive this ranking'? Once that is known, the user knows which node to be very careful on completing with information, since small changes to information on that node are very influential. For the other nodes, the input of data and quality control thereof can be cursory. (Inputting data is time-consuming and tiring.) I.e., a cursory data set can be input into ten nodes, and the user can be very careful with gathering and inputting data into the other five nodes which are identified as the most important/influential.

Consider another example of ranking the top 20 college football teams. They will be scored against offense, defense, alumni, and the like. The user wants to know which node will most affect the ranking of the teams. Did the alumni have the largest effect on the ranking of the teams, or is it offense or defense, or coaching budget? The system and method described below can isolate which are the most influential to the ranking of the football teams. The issue is, which factors influence the ranking of the teams; the issue is not how the teams are ranked. In operation of this model, someone is following the teams in general and wants to know the main influencer to the ranking of the teams—offense, defense, budget, coaching talent, etc.—so any change can be analyzed to determine which change is most likely to affect ranking. For example, coaching talent might be determined to be most influential in the ranking, and thus, whichever team had a change in coaching talent is most likely to go up or down vs. a team with a change in offense is less likely to affect the ranking

DEFINITIONS

The following definition section includes some of the terms which are used herein as having the definitions set out herein, since sometimes it is helpful to have a definition section. However, this definitions section does not exhaustively list all of the terms defined herein. The remainder of the specification may include definitions for additional terms.

The designations in the following list are defined and expressly used herein as would be understood to one of skill in the ANP art, and not according to a general dictionary, whether singular or plural: “priority,” “node”, “importance” (or “important”), “sensitivity,” “pairwise comparison”, “ANP ratings” (or “ratings”), “ANP client data”, “priority vectors”, “judgment of priority,” “ANP alternatives” (or “alternatives”), “criteria”, “weight,” “cluster,” “local,” “global,” “synthesize.” The list in this paragraph is not exhaustive and does not imply that a term not on this list can be interpreted according to a general dictionary instead of according to an understood ANP meaning. Some of these are further explained below for the reader's convenience.

“Pairwise comparison.” The point of a pairwise comparison set is to arrive at the priorities of a group of things. These things may be criteria (so-called “alternatives” in the traditional ANP sense), or ratings scales. In a classic example of doing pairwise comparisons, one can answer the question, “how many times better is X than Y” for all X and Y being compared.

“ANP Ratings.” If one thinks of a single column of the conventional ANP's rating system as conventionally represented in a matrix, its point is to assign ideal priorities to the alternatives (with respect to a criteria). The ratings prioritize alternatives in a set of alternatives. In a classic example of doing ANP ratings, one first sets up ratings words like “High”, “Medium” and “Low”, and gives scores to those words; then each of the things being rated is assigned a “High”, “Medium” or “Low.”

“ANP client data.” ANP client data are data that represent real world values. For example, in a decision about an automobile purchase, ANP client data could be miles per gallon, number of passengers, headroom, cubic size of storage, etc.

The term “analytic network process” (“ANP”) model, sometimes referred to as an ANP network model, an ANP network or similar, is defined herein to refer to a form of an analytic hierarchy process (AHP) in which values for higher level elements are affected by lower level elements and take the dependency of the lower level elements into account; further in the ANP model, the importance of the criteria determines the importance of the alternatives (as in an AHP); the importance of the alternatives themselves determines the importance of the criteria; further, the ANP model additionally has influence flowing between non-downward elements (in comparison to a conventional AHP model, in which influence flows only downwards); further the ANP model is a network, that is not a top-down-tree, of priorities and alternative choices. The terms “criteria” and “alternatives” are understood to those of skill in the AHP art. An ANP is further discussed in, e.g., Saaty, T. L. (2001) Decision Making with Dependence and Feedback: the Analytic Network Process, 2^(nd) edition.

The term “ANP weighted supermatrix” is defined as the supermatrix which is created from the ANP model, and which has been weighted, in accordance with ANP theory, and variations, extensions, and/or evolutions of such ANP theory. The ANP supermatrix is understood to be represented in rows and columns.

END OF DEFINITIONS 1. Introduction

Given an ANP model, it is natural to wonder how sensitive the scores of the alternatives are to numerical changes in the model. Although, changes in the scores of the alternatives are often less important than changes in the rankings (That is a numerical change in the model that changes all of the scores a lot, but leaves the rankings the same is, in some sense, far less interesting than a numerical change which drops the number one alternative to the number four position.)

Rather than speaking about broad “numerical changes” influencing alternative rankings we can consider the effects a particular node has on the rankings (how a node can impact the rankings of alternatives will be explained shortly, for now think of it like tree sensitivity). We could look at the effects each node has on the rankings of the alternatives, and then score the nodes based on this information (a higher score for a node means it impacts the rankings of the alternatives more). With information like this, we can rank the nodes in the model from most influential to the rankings of the alternatives to least influential. We present a method to accomplish this goal using ANP row sensitivity (defined in part III) as the basis of the calculations.

1.1 The Concept

The basic idea utilized herein is the following.

Concept (ANP Rank Influence). For each node we consider how much we would have to change the importance of that node to induce a change in the rankings of the alternatives. Nodes that require a small change of importance to create a rank change score higher than nodes that require a large change (if we need only barely change the importance of a node to get a change in the rankings of the alternatives that node is very influential to the rankings).

There are a few subtleties involved in implementing this concept; however, the above concept is the driving force behind the calculations we do. Implementing this concept requires a few steps.

-   -   1. We need to be able to change the importance of a node in a         way that is compatible with the ANP structure of the model. We         use ANP row sensitivity to accomplish this (see Part II for a         full explanation of row sensitivity, in addition we provide a         brief review in the following section).     -   2. We need to consider changing the importance of a node upward         as well as downward.     -   3. We need a method for scoring a node's importance based upon         how far its importance had to move to produce a rank change of         the alternatives. There is a subtlety involved with reconciling         the impact of moving a node's importance upward versus downward         as moving downward simply has less of a numerical impact than         moving upward.

1.2 Review ANP Row Sensitivity

The following is a brief review of the concepts involved in Part II. The purpose of ANP row sensitivity is to change all of the numerical information for a given node in a way that is consistent with the ANP structure, and recalculate the alternative values (much as tree sensitivity works). We do this by having a single parameter p that is between zero and one, which represents the importance of the given node. There is a parameter value p₀ (called the fixed point) which represents returning the node values to the original weights. For parameter values larger than p₀ the importance of the node goes up, and for parameter values less than p₀ the importance of the node goes down. Once the parameter is set, this updates values in the weighted supermatrix (although it can also be done with the unscaled supermatrix, working by clusters instead) and resynthesizes.

2 ANP Rank Influence

We shall begin the discussion of ANP rank influence by first restating the algorithm in a more technical fashion (and then proceed to the official technical definition). Fix an ANP model (a single level of it) let Wbe its weighted supermatrix (of dimensions n×n) and 1≦r≦n be a row of W. Let W(p) be a family of row perturbations for row r of W with p₀ as the fixed point (see Part II) (for instance, W(p) could be F_(W,r,p) ₀ (p) that is defined in Part II). The algorithm comprises searching for the first p⁺ above p₀ where a rank change occurs and p⁻ below p₀ where rank change happens. Using p⁺ and p⁻ we construct a number that tells us the upper and lower rank influence of row r in W.

2.1 Technical Definition

In order to state things precisely we need a few definitions.

Definition 1 (Ranking) Let A be an ANP model with a alternatives ordered. We use the following notation for standard calculated values of the model.

S_(A,i)=synthesized score for alternative i

r_(A,i)=ranking of alternative i where 1=best, 2=second best, etc.

Definition 2 (Family of ANP models induced by row perturbations). Let A be an ANP model, W be the weighted supermatrix of a single level of it (of dimensions n×n) and let W(p) be a family of row perturbations of row 1≦r≦n of W. We can think of this as inducing a family of ANP models, which we denote by A(p).

Definition 3 (Upper and lower rank change parameter). Let A be an ANP model, W be the weighted supermatrix of a single level of it (of dimensions n×n) and let W(p) be a family of row perturbations of row 1≦r≦n of W. Let A(p) be the induced family of ANP models. We define the upper rank change parameter p⁺ _(A,W,r) and lower rank change parameter p⁻ _(A,W,r) by the following formulas. p ⁺ _(A,W,r)=inf{p≧p ₀ |A(p) has a different ranking than A} p ⁻ _(A,W,r)=sup{p≦p ₀ |A(p) has a different ranking than A}

When the context is clear we abbreviate the upper and lower rank change parameters as p⁺ and p⁻. If we only wish to emphasize the row which we are analyzing we write p_(r) ⁺ and p_(r) ⁻ for p⁺ _(A,W,r) and p⁻ _(A,W,r), respectively.

Note 1. By continuity considerations the above inf and sup are simply maximum/minimums if there is no ties in the scores of the alternatives of A. However when there are ties continuity alone does not suffice to reduce the inf and sup to maximum/minimum.

2.2 Rank Influence Score

Using the information of p⁺ _(r) and p⁻ _(r) we can construct an upper and lower rank influence score that tells us how quickly rank changes occur in row r. The larger this score the more rank influence row r has.

Definition 4 (Rank Influence Score). Let A be an ANP model, W be the weighted supermatrix of a single level of it (of dimensions n×n) and let W(p) be a family of row perturbations of row 1≦r≦n of W. Let A(p) be the induced family of ANP models. We define the following rank influence scores. The upper rank influence score rki⁺ _(A,W,r) (or simply rki⁺ when no confusion would arise) is defined by

${rki}_{A,W,r}^{+} = {\frac{1 - p_{A,W,r}^{+}}{1 - p_{0}}.}$

Likewise the lower rank influence score rki⁻ _(A,W,r) is defined by

${rki}_{A,W,r}^{-} = {\frac{1 - p_{A,W,r}^{-}}{1 - p_{0}}.}$

Note 2. Both rki⁺ and rki⁻ are between zero and one. If no switch happens on the upper side we have rki⁺=0 (respectively for rki⁻). Likewise if p⁺=p₀ then rki⁺=1 (likewise for rki⁻). Lastly notice that rki⁺ is a decreasing function of p⁺ and rki⁻ is an increasing function of p. This means that if no rank change happens no matter how much we change p in the upward direction, we get p⁺=0 (likewise for the downward direction). The closer the upper (or lower) rank change parameter is to p₀, the closer the score is to 1 as well.

There is a natural problem with the above definition. Namely for any row we would like to have a single rank influence score, not an upper and lower influence score. We could simply take the larger of these two scores. However, there is a problem with that approach which we outline and address in the following section.

2.3 Adjusted Lower Rank Influence Score

There is an inherent difficulty (or unfairness) in directly comparing lower and upper rank scores to obtain a total rank influence score. The problem is that by its nature changing the parameter p upwards has a much larger influence on the supermatrix than changing downwards (below p₀). In the extreme (with p near 1) the supermatrix has nearly 1's in the r^(th) row with nearly zeros in all other rows. However in the extreme downwards, the r^(th) row is made nearly zero, while all other rows get bumped up a bit.

To address this problem we need to adjust the lower rank influence score upwards in some fashion. There are many ways to do this of course. We must be careful though. The upper rank influence score is between 0 and 1, and we would like to keep the adjusted lower rank influence score in this range as well. A natural way to do this is by taking an appropriate root of the (raw) lower rank influence score to arrive at an adjusted rank influence score, which is what we propose here.

Definition 5 (Root adjusted lower rank influence score). Let rki⁻ be the lower rank influence score. We define the m^(th) root adjusted lower rank influence score to be rki ^(−,m)=^(m)√{square root over (rki ⁻)}.

Note 3. From trial and error on various models, the best values form appear to be either 2 or 3. That is, using either the square root or cube root appears to make lower and upper rank influences comparable (for most models). It would be nice to find a common root to use for all models to adjust the lower rank influence score by. The most likely candidate based upon the previous remarks would be the e^(th) root. More work would need to be done to verify this though.

2.4 Algorithms Finding Upper and Lower Rank Change Parameter

Determining the rank influence score (scores) requires us to find the upper and lower rank change parameter value. Thus we must search for the smallest parameter value which causes a rank change. (We should mention here that if we start with some alternatives in a tie, it is very possible that you would need to take an inf or sup instead of maximum or minimum. In those cases the lower or upper rank change parameter may in fact be p₀. This is not a problem with the theory as advanced here. However it should be noted as a degenerate (or at least a strange) case.) There are several algorithms available for such a search, of which we discuss a few below.

2.4.1 Bisection Method Approach

The bisection method is a standard method for finding roots of a function (or in general elements of a set that satisfy a certain criteria). The method consists of dividing the set you are searching in half (if it is a finite set this is easy, in our case we have intervals we are searching, and it is easy to divide the set in our case as well), then choosing one half to restrict our search to. Then we continue the process of dividing. Techniques are known for performing bisection methods.

In our case to find the lower rank change parameter (the same idea applies to the upper rank change parameter) we start with the interval [0,p₀] through which we will search for our lower rank change parameter. We start with the alternatives scored as normal with the parameter value of p₀. The algorithm consists of several steps. The guiding idea is to restrict to intervals we know rank change occurs within.

1. In order to start the algorithm we need to know that our starting interval has a rank change occur within it. The test is straightforward, we check to see if a rank change happens at parameter value 0 (the other end of our interval). If that does not cause rank change, we have two choices. Either we can give up (assuming that if no rank change has happened at the end, there was probably no rank change in between). Or we can use another algorithm to search for some parameter value which causes rank change (such as the brute force method listed below) and return to the bisection algorithm with the parameter value causing rank change as our new end point for the search interval.

2. Once we have established that our starting interval's end points have different rankings, we choose the midpoint. If the midpoint does not cause a rank change (from the starting rankings), we choose the lower half of the interval to search within. If the midpoint does cause a rank change we choose the upper half of the interval to narrow our search to.

3. Repeat the previous step until we narrow the interval down to the desired size. Every step reduces the size by a factor of two. So after n steps, we know a rank change occurs somewhere within an interval of size p₀/2^(n), with the upper end point of that interval not causing a rank change and the lower end causing one.

There is, of course, a problem with this. Our lower rank change parameter is the supremum of the set of parameter values causing rank change. We may not have found the largest interval to search within with this algorithm. However, the computational needs of this algorithm are very small to get to a large accuracy.

2.4.2 Brute Force Search Approach

As we did for the bisection method, we will describe this approach for the lower rank change parameter (the algorithm is nearly identical for finding the upper rank change parameter). The idea is very simple. We start by defining a step Δp that we use throughout the search. We then start at p₀ and decrease by Δp and see if a rank change has occurred. If there is not a rank change we decrease by Δp again, and continue until we find a rank change.

This method has a much better chance of finding a value close to the actual lower rank change value (assuming Δp is sufficiently small). However, if the lower rank change parameter is much smaller than p₀ we would need to search for quite a while to find it. Also we have a fixed accuracy for the calculation in the brute force approach (namely Δp) whereas the bisection method gets more accurate each step.

2.4.3 Hybrid Approach

A hybrid approach marries the best of both of the bisection and brute force methods. As mentioned in the bisection method discussion we can use the brute force method to find the initial end point of our search interval if 0 is not a rank change point.

However we could use the brute force algorithm to find the initial end point even assuming 0 is a rank change parameter value. Additionally we can find a new starting point for the other side of the interval. The brute force algorithm will find a parameter value p′ that causes a rank change, and also we know that p′+Δp does not cause a rank change. So we have limited our search region to the interval [p′,p′+Δp]. By doing this we have a better chance of finding the largest rank change parameter value.

We can tweak the computational time needed by making Δp larger (and then allowing the bisection method to hone in and reduce the inaccuracy). This growing of Δp does come at a cost, namely the possibility of missing a larger value for rank change (and thus giving us an incorrect value for the lower rank change parameter). The possibility of this happening is particularly low since it would require a “wiggling” behavior of the alternative scores versus the parameter value on a small scale (this has not been seen in any working model, though certainly must be considered as a possibility).

3 Examples

We present test results obtained from a new system (based on Super Decisions) performing the rank change algorithm. For all of the data presented the new system based on Super Decisions used the hybrid algorithm for finding the upper and lower rank change parameter. Please note that the new system based on Super Decisions, at the time of writing this paper, does not join lower and upper rank influence scores into a single score. Instead results are returned about both upper and lower rank influence.

3.1 4Node2.mod

This model is the model as used in the examples of Part III. It is a model with two clusters (a criteria cluster and alternatives cluster) each of which contain two nodes (two criteria “A” and “B” and two alternatives “1” and “2”). All nodes are connected to one another with pairwise comparison data inputted.

For this model both the hybrid and the bisection approach arrived at the same results. Results for the rank influence of the alternatives are included because the results for the alternatives have something useful to show us. Note that the results in Table 1 have been organized from highest to lowest score.

TABLE 1 Node Param Score Raw Score Alt 1 Score Alt 2 Score Original 0.5000 0.0000 0.0000 0.3941 0.6059 Values 1:upper 0.5469 0.9043 0.9043 0.5004 0.4996 2:lower 0.3594 0.8958 0.7188 0.5002 0.4998 A:upper 0.6560 0.6816 0.6816 0.5002 0.4998 B:lower 0.0171 0.3245 0.0342 0.5001 0.4999 B:upper 0.9900 0.0000 0.0000 0.1045 0.8955 2:upper 0.9900 0.0000 0.0000 0.0031 0.9969 A:lower 0.0000 0.0000 0.0000 0.2738 0.7262 1:lower 0.0000 0.0000 0.0000 0.0000 1.0000

Before we discuss what this information means, we should understand the columns of the above data. Each row represents information about lower/upper rank change for a given node.

Node

The node whose rank change information we are exploring. The :lower means we are looking at lower rank change information for that node, and :upper means we are looking at upper rank change information for that node.

Param

The actual parameter value we found to be the lowest/highest (depending if we are doing upper or lower rank change analysis respectively) causing rank change.

Score

This is the upper rank influence score if we are doing upper rank influence analysis. It uses the root adjusted lower rank influence score if we are doing lower rank influence analysis, using the cube root.

Raw Score

This is simply the rank influence score (i.e. we do not do any adjustments of the lower rank influence score for this column).

Alt 1 Score

The normalized score of alternative “1” when the parameter value has been changed to the value given in the param column.

Alt 2 Score

The normalized score of alternative “2” when the parameter value has been changed to the value given in the param column.

With this information in hand we can make a few observations.

(1) The top two rank influencers are the alternatives (which makes sense). However it is only “1:upper” and “2:lower” that have influence (that is “1:lower” and “2:upper” have no rank influence. This makes sense as originally alternative “1” scored 0.3941 and alternative “2” scored 0.6059. That is alternative “2” ranked best and “1” ranked second. Increasing the importance of “1” of course makes a difference to the rankings (whereas decreasing the importance of “/” has no effect on the rankings). Likewise, in reverse, for “2”.

(2) The ranking of most influential to least remains the same whether we use the adjusted lower rank influence score, or the simple lower rank influence score. This is not often the case as we will see shortly.

(3) Criteria “A” is the most influential non-alternative (although it is only in the upper direction that it is the most influential). From Part III the influence analysis shows that “B” is actually the most influential (in terms of raw numerical change). We now can see that, although “B” changes the alternative scores the most numerically, it does not effect the ranking at all in the process. It is criteria “A” that is the most rank influential.

3.2 BigBurger.mod

This model is included simply to show that rank influence will not always tell us useful information. The only nodes in the BigBurger model that have rank influence are the alternatives. That is, no non-alternative nodes cause a rank change.

3.3 DiLeo&Tucker Beer Market Share

This is a model taken from the Saaty's class on ANP. The model is designed to predict market share of various beer manufacturers. Since we have previously described the columns of the data, we simply need to note in Table 2 that last four columns are the normalized scores of the alternatives at the various parameter values.

TABLE 2 Node Param Score Raw Busch Coors Other Miller Original Values 0.500 0.000 0.000 0.434 0.1612 0.2013 0.2032 Quality: high 0.502 0.996 0.996 0.432 0.1617 0.2031 0.2027 Customers: high 0.503 0.994 0.994 0.433 0.1616 0.2029 0.2028 Ad Spend: low 0.491 0.994 0.981 0.431 0.1621 0.2037 0.2036 Availability: low 0.476 0.984 0.952 0.435 0.1610 0.2022 0.2022 Price: high 0.508 0.983 0.983 0.431 0.1622 0.2036 0.2036 Creat Ads: low 0.448 0.964 0.896 0.433 0.1616 0.2028 0.2028 Appeal: high 0.518 0.963 0.963 0.433 0.1611 0.2031 0.2031 Brand Rec: low 0.440 0.959 0.881 0.433 0.1615 0.2028 0.2028 Style: high 0.521 0.957 0.957 0.433 0.1606 0.2035 0.2035 Freq Ads: low 0.325 0.866 0.650 0.432 0.1617 0.2031 0.2031 Availability: high 0.595 0.807 0.807 0.428 0.1659 0.1658 0.2400 Appeal: low 0.202 0.739 0.403 0.434 0.1634 0.2015 0.2014

This model is very rank sensitive, and the reason can be seen by looking at the original values. The alternatives “Other” and “Miller” scored very close to each other originally. This is our first instance of rankings according to rank influence differing whether we use the adjusted lower rank influence score, or the raw lower rank score. The above has been sorted using the score (using the cube root adjusted lower rank influence score). If we sorted on raw rank influence we see different orderings in some cases.

It is also interesting to compare these values to the influence calculations described in Part III. In the above Table 2 we have the top twelve scoring alternatives, below in Table 3 we have the top twelve scoring alternatives using influence analysis with upper value of 0.9 and lower value of 0.1. The table is sorted by rank change first then distance.

TABLE 3 Node Param Dist Rank Ch Busch Coors Other Miller Original Values 0.5 0.0000 0 0.434 0.161 0.201 0.203 Quality: high 0.9 1.1573 8 0.165 0.239 0.462 0.134 Customers: high 0.9 1.1573 6 0.242 0.213 0.388 0.156 Price: high 0.9 1.1573 4 0.270 0.210 0.293 0.227 Ad Spend: low 0.1 1.1573 4 0.268 0.200 0.303 0.229 Availability: high  0.9 0.8843 2 0.395 0.184 0.072 0.348 Ad Spend: high 0.9 0.7420 2 0.614 0.113 0.073 0.200 Appeal: high 0.9 0.4317 2 0.383 0.166 0.254 0.197 Style: high 0.9 0.3080 2 0.400 0.151 0.242 0.207 Availability: low 0.1 0.0931 2 0.439 0.158 0.216 0.186 Creat Ads: low 0.1 0.0835 2 0.423 0.164 0.213 0.200 Brand rec: low 0.1 0.0724 2 0.425 0.164 0.211 0.200 Freq Ads: low 0.1 0.0319 2 0.429 0.162 0.205 0.203

It is interesting to note that the first three nodes match up perfectly between influence analysis and rank influence analysis. Many other remain the same.

The first change is that “Price” moves up a bit. The node “Ad Spending” (the upper values of it) does not appear in rank influence, yet does make an appearance in influence (meaning that moving “Ad Spending” upwards does not influence the ranking as much as it affects the raw numbers).

Referring now to FIG. 1, a block diagram illustrating a system for ANP rank influence analysis will be discussed and described. In the illustrated embodiment, there is provided a controller 107, with an ANP rank influence analysis unit 103. The ANP rank influence analysis unit 103 can access an ANP storage memory 105, in order analyze the ANP model in the ANP storage memory 105. Users can interact via an output unit 101 b and/or an input unit 101 d with the ANP rank influence analysis unit 103. Also, users can interact via an input unit 101 d with the ANP model stored in the ANP storage memory 105, for example where votes for the ANP model (which can represent votes) are input via the input unit 101 d. The output unit 101 b and/or input unit 101 d can be remote or local.

Referring now to FIG. 2, a diagram illustrating a simplified representation of an ANP model will be discussed and described. The illustration is simplified for ease of discussion. In the ANP model, there are conventionally provided control criteria that are benefits, costs, opportunities, and risks (commonly abbreviated BOCR). At the top of the ANP model 200, there is provided an ANP model goal 201, benefits 203 a and opportunities 203 b. (The usual costs and risks are not shown.) The benefits 203 a is a node that includes a one way directional link from the benefits 203 a node to the social benefits node 205 a and the political benefits node 205 b. The opportunities 203 b is a node that includes a one way directional link from the opportunities 203 b node to the social opportunities node 205 c and the political opportunities node 205 d. The political benefits node 205 b includes a one way directional connection to the benefits node 203 a and the opportunities node 203 b. Each of the social benefits node 205 a, the political benefits node 205 b, the social opportunities node 205 c and the political opportunities node 205 d includes a separate one-way directional connection to alternative 1 211 a and alternative 2 211 c.

In a conventional ANP model, the connection defines how important the destination node is to the source node. Hence, a connection is directional, that is, it has a from direction and a to direction. For example, a connection from the conventional ANP model goal 201 to the benefits node 203 a means that the user can define how important benefits are to the goal.

One of skill in this art will know that the ANP model can be represented as a matrix (or series of matrices), where a node is represented as a row in the matrix.

Referring now to FIG. 3, a block diagram illustrating portions of an exemplary computer will be discussed and described. The computer 301 may include one or more controllers 303 having an optional communication port 339 for communication with an external device (not illustrated), a processor 309, and a memory 311; a display 305, and/or a user input device 307, e.g., a keyboard (as illustrated), trackball, mouse, or known voting device. Many of the other elements of a computer are omitted but will be well understood to one of skill in the art.

The processor 309 may comprise one or more microprocessors and/or one or more digital signal processors. The memory 311 may be coupled to the processor 309 and may comprise a read-only memory (ROM), a random-access memory (RAM), a programmable ROM (PROM), and/or an electrically erasable read-only memory (EEPROM). The memory 311 may include multiple memory locations for storing, among other things, an operating system, data and variables 313 for programs executed by the processor 309; computer programs for causing the processor to operate in connection with various functions such as to store 315 an ANP weighted supermatrix representing an ANP model into memory, to analyze 317 ANP rank influence, to input 319 various data into the ANP model, and/or other processing 321; the ANP storage memory 323 in which the ANP weighted supermatrix is stored; and a database 325 for other information used by the processor 309. The computer programs may be stored, for example, in ROM or PROM and may direct the processor 309 in controlling the operation of the computer 301.

The user may invoke functions accessible through the user input device 307. The user input device 307 may comprise one or more of various known input devices, such as a keypad, a computer mouse, a touchpad, a touch screen, a trackball, a keyboard and/or a button device configured to register votes. Responsive to signaling received from the user input device 307, in accordance with instructions stored in memory 311, or automatically upon receipt of certain information via the communication port 339, the processor 309 may direct information in storage or information received by the user input device to be processed by the instructions stored in memory 311.

The display 305 may present information to the user by way of a text and/or image display 305 upon which information may be displayed. The display 305 may present information to the user by way of an available liquid crystal display (LCD), plasma display, video projector, light emitting diode (LED) or organic LED display, cathode ray tube, or other visual display; and/or by way of a conventional audible device (such as a speaker, not illustrated) for playing out audible messages.

The processor 309 can be programmed to store 315 an ANP weighted supermatrix representing an ANP model into memory. Before storing the ANP weighted supermatrix, values in the ANP weighted supermatrix can be obtained from an ANP model, for example, by inputting pairwise comparisons and creating an ANP weighted supermatrix therefrom, through various known techniques. An ANP weighted supermatrix representing the ANP model which is created can be generated, and the ANP supermatrix and/or the ANP model can be stored in memory. The memory can be local as illustrated (e.g., ANP storage memory with ANP weighted supermatrix), or can be remote if preferred.

The processor 309 can be programmed to analyze 317 ANP rank influence. This is discussed elsewhere in this application in detail, and will not be repeated here.

The processor 309 can be programmed to interact with the user so as to input 319 new or modified pairwise comparisons, ANP ratings, and/or ANP client data, and transform the data into priority vectors and store into the ANP model. As with traditional ANP, alternatives can be pairwise compared. The data which is input can be transformed into priority vectors, as with traditional ANP, and matrix transformations can be prepared. The result can be stored into the ANP, such as the ANP storage memory 323 with ANP weighted supermatrix in the memory 311.

Optionally, other components may be incorporated in the computer 301 to produce other actions. For example, a user can interface with the computer 301, via a known user interface such as OUTLOOK software, WINDOWS software, and/or other commercially available interfaces. Further, the computer 301 can send and receive transmissions via known networking applications operating with the communication port 339 connected to a network, for example, a local area network, intranet, or the Internet and support software.

It should be understood that various embodiments are described herein in connection with logical groupings of programming of functions. One or more embodiments may omit one or more of these logical groupings. Likewise, in one or more embodiments, functions may be grouped differently, combined, or augmented. For example, in one or more embodiments, the synthesizer can be omitted. In addition, some of these functions may be performed predominantly or entirely on one or more remote computers (not illustrated); and therefore such functions can be reduced or omitted from the processor 309 and distributed to the remote computer. Similarly, the present description may describe various databases or collections of data and information. One or more embodiments can provide that databases or collections of data and information can be distributed, combined, or augmented, or provided locally (as illustrated) and/or remotely (not illustrated).

The ANP storage memory 323 is illustrated as being part of memory 311 stored locally on the controller 303. It will be appreciated that the ANP storage memory 323 can be stored remotely, for example, accessed via the communication port 339 or similar.

The computer 301 can include one or more of the following, not illustrated: a floppy disk drive, an optical drive, a hard disk drive, a removable USB drive, and/or a CD ROM or digital video/versatile disk, which can be internal or external. The number and type of drives can vary, as is typical with different configurations, and may be omitted. Instructions that are executed by the processor 309 and/or an ANP model can be obtained, for example, from the drive, via the communication port 339, or via the memory 311.

The term “computer-readable medium” is intended to cover only statutory computer memory devices, that is, non-transitory devices.

The above is sometimes described in terms of a single user, for ease of understanding and illustration. However, it is understood that multiple users can be accommodated in various embodiments. For example, multiple users each can input pairwise comparisons.

Furthermore, the devices of interest may include, without being exhaustive, general purpose computers, specially programmed special purpose computers, personal computers, distributed computer systems, calculators, handheld computers, keypads, laptop/notebook computers, mini computers, mainframes, super computers, personal digital assistants, communication devices, any of which can be referred to as a “computer”, as well as networked combinations of the same, and the like, although other examples are possible as will be appreciated by one of skill in the art, any of which can be referred to as a “computer-implemented system.”

One or more embodiments may rely on the integration of various components including, as appropriate and/or if desired, hardware and software servers, database engines, and/or other content providers. One or more embodiments may be connected over a network, for example the Internet, an intranet, a wide area network (WAN), a local area network (LAN), or even on a single computer system. Moreover, portions can be distributed over one or more computers, and some functions may be distributed to other hardware, in accordance with one or more embodiments.

Any presently available or future developed computer software language and/or hardware components can be employed in various embodiments. For example, at least some of the functionality discussed above could be implemented using C, C++, Java or any assembly language appropriate in view of the processor being used.

One or more embodiments may include a process and/or steps. Where steps are indicated, they may be performed in any order, unless expressly and necessarily limited to a particular order. Steps that are not so limited may be performed in any order.

Referring now made to FIG. 4, an example implementation for illustrating rank influence analysis will be discussed and described. This model is designed to evaluate the top four NCAA football teams as of the ranking on Sep. 22, 2011. The model is not intended to be exhaustive, rather its purpose is to illustrate the usefulness of rank influence. FIG. 4 reproduces an image of the ANP model 400 on a user interface as seen in Super Decisions. In the model there is an Alternatives cluster 409 for the teams we are ranking, as well as the following criteria clusters 401, 403, 405, 407:

-   -   Overall: This cluster 401 contains the large influences on a         team's ranking, Offense, Defense, and Coaching. Teams are not         compared with respect to these criteria. The values for these         criteria come from the statistically oriented criteria they         connect to.     -   Offensive Stats: The criteria for basic offensive statistics are         placed in this cluster 405, and teams are scored with respect to         these based upon those statistics.     -   Defensive Stats: The criteria for basic defensive statistics are         placed in this cluster 407, and teams are scored with respect to         these.     -   Team Stats: The team wide statistics are placed in this cluster         403 and the teams are scored with respect to these.

After all of the pairwise, ratings, and direct data is inputted into this model we have the following synthesized scores for the teams (numbers are quoted in their normalized form, that is, all the scores add up to 1.0), shown in Table 4.

Table 4:

Team Score Rank LSU 0.247 3 Oklahoma 0.286 1 Alabama 0.271 2 Ball State 0.195 4

The question is which of the criteria is most influential to the ranking of the teams. The rank influence analysis tells us that information, and the following Table 5 describes the rank influence of the most influential nodes, in descending order.

TABLE 5 Criteria Parameter Rank Influence Defensive Sacks/gm:upper 0.530395 0.937969 Interceptions/gm:upper 0.542684 0.912891 Defensive Fumble Gained/gm:upper 0.547229 0.903613 Pass yd/gm:upper 0.589865 0.816602 Sacks/gm:upper 0.596947 0.802148 Pass yd allowed/gm:upper 0.603694 0.788379 Points scored/gm:upper 0.609102 0.777344 Time of possession:upper 0.622299 0.75041 Defensive Interceptions/gm:upper 0.6372 0.72

The first column is the criteria being analyzed (the :upper means we are looking at rank influence moving the priority of the criteria upwards). The second column is the parameter value that caused the first change in the rankings, and the third column is the rank influence score. From these calculations, for this model we see that the defensive sacks per game is the most influential to changing the rankings. Next most influential is interceptions per game (that is the offense turning the ball over through an interception). In addition, rank influence shows that the following criteria have no effect of the rankings:

Fumbles lost/gm

Coaching

Defense

Points allowed

Referring now to FIG. 5, a flow chart illustrating a procedure 501 to analyze rank influence of an ANP model will be discussed and described. The procedure can advantageously be implemented on, for example, a processor of a controller, described in connection with FIG. 3 or other apparatus appropriately arranged.

As illustrated in FIG. 5, the procedure 501 will store 503 an ANP model with feedback connections in place, wherein the ANP model has stored therein at least has preliminary input of data for the ANP model.

Then, the procedure 501 can interact with one or more users to input 505 revised or new pairwise comparisons, ANP ratings, and/or ANP client data; to transform the input into priority vectors; and to store the priority vectors into the ANP model. This can be done in accordance with known techniques for modifying data in an ANP, such as by interacting with a user. The user interface side of inputting pairwise comparisons, ratings, or client data can be performed according to known techniques. For example, the process 501 can query the user to input, “with respect to opportunities, which is more important: social or political?” to input values of a pairwise comparison of the social and political opportunities nodes. Also, the process 501 can transform the input values into priority vectors in accordance with known techniques. Further, the process 501 can store the new or modified input values and the priority vectors into the ANP model.

Then, the procedure 501 can perform 507 rank influence analysis to determine one or more nodes among the nodes within the ANP model in which a smallest upward and/or downward change in importance will change any rank among the nodes in the ANP model. This can include basing the rank of each node on an amount of influence of each said node on rankings of alternatives in the ANP model. This can include finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model.

Upper and lower rank changes are determined because the changes are usually not necessarily symmetrical in an ANP. This is where the natural fixed point comes into play. Consider, for example, a node in a large model with 10 nodes. If every node is equal, each node has about 1/10^(th) of the weight. If the node is moved upward from 0.1, then it gets proportionately more and every other node goes much lower. So, the upper rank change, that is when moving the node's weight upward, is different from the lower rank change, that is when moving the node's weight downward.

Then, the procedure 501 can determine 509 a rank score for the nodes in the ANP model based on an amount of upward and/or downward change in importance that causes a change in rank among the nodes.

Then, the procedure 501 can indicate 511 one or more of the nodes determined to have the most influence on the rank among the nodes, for example, the nodes which are ranked more highly to indicate the node with the smallest change that will cause a change in rank.

Then, the procedure 501 can prompt 513 the user to input new or refined or additional data for the one or more of the nodes determined to have the most influence, while canceling a requirement to input new, refined, or additional data for the remainder of the nodes in the ANP model. One of the points for determining the rank influence is to be able to focus user efforts on refining data for the nodes with the most influence, whereas the preliminary input of data initially stored in the ANP model for the nodes which do not have the most influence does not need to be refined.

Optionally, if a user desires to input additional data for the nodes which do not have the most influence, the user interact with the procedure 501 to input 515 revised or new pairwise comparisons, ANP ratings, and/or ANP client data; transform the input into priority vectors; and store the priority vectors into the ANP model.

Note, in some implementations, the procedure 501 can omit prompting a user to input 513 refined data for the nodes determined to have the most influence. For example, in some situations it may be assumed that the ANP model already has refined data stored therein.

Also, the procedure 501 can omit interacting with the user to input 505, 515 revised or new pairwise comparisons, ANP ratings, and/or ANP client data, and transforming and storing the input.

Consequently, the system or method can be used to perform an initial run through on the data, then highlight the most important nodes such as the top five or top ten nodes. The user would be prompted for data input for those nodes. The user can then interact with the system or method to fine tune the data for those nodes, and skip the other nodes. This may be particularly helpful during model construction.

As another example, the system or method can be used to provide a list of nodes with a value beside them showing magnitude of rank influence of each node. For example, this can be provided as a graphical display, such as in response to a drop-down request, perhaps along with other influence illustrations. This can be particularly helpful for post model construction analysis.

Part Ii: Row Sensitivity Analysis

Sensitivity analysis in ANP has several difficulties. The goal of sensitivity analysis is to discover how changes in the numerical information in an ANP model affect the scores for the model's alternatives. The numerical data involved could be information directly supplied to the model, such as pairwise data. On the other hand we could also want to analyze sensitivity to calculated data, such as local priorities, or global priorities. These methods do indeed show us certain levels of sensitivity. However, for the vast majority of single level ANP models, they either report useless information (tweaking global priorities is only useful in multi-level models at best), or no sensitivity (a single pairwise comparison has no effect in a well connected ANP model, likewise a single local priority has no effect in a well connected ANP model). These do little better for multi-level models.

The problem is how to perform acceptable AHP tree-type sensitivity measurements in the ANP network setting. We want to be able to have an analysis that will come up with a result which is similar to AHP tree sensitivity but in an ANP context. We want to be able to analyze the ANP network to see how influential the nodes are or how sensitive our results are to the nodes. There are pre-existing methods and systems for performing sensitivity analysis in an ANP. These ideas, however, are still lacking

The systems and methods herein concern a new type of sensitivity analysis that gives rise to useful sensitivity in ANP modes, even single level ANP models, where other methods have failed. We will use the terminology “row sensitivity” for this new kind of analysis. We will show that, if we accept certain axioms about preserving ANP structure, row sensitivity as outlined here, is a kind of calculation we can perform. It appears that any other analysis will disrupt the basic structure of the model, rendering the results less meaningful. We feel obliged to note that, although we speak throughout here of single level ANP models, row sensitivity is equally useful in multiple level ANP models. In fact, it serves, in many respects, as a superior replacement to global priorities sensitivity analysis, in that the former can preserve the overall structure of the model in a way that the latter cannot.

1.1 Sensitivity in AHP Trees

By way of introduction, let us review the basic idea and result of standard sensitivity analysis in the case of AHP trees. Although the AHP tree case does not show us the way to proceed, it does show us the kind of information we would like to glean from sensitivity.

In typical AHP tree sensitivity, we take the local weights for the collection of criteria under a common parent, and drag up or down a particular criteria's weight. Since we are dealing with a tree, a criteria's local weight and global weight are essentially the same (a simple rescaling is the only change that happens to go from local to global). By changing said local weight (or weights) we get new local priorities for the criteria in question, and re-synthesize to get new scores for our alternatives. By dragging a single criteria's priority towards one or towards zero, we get an idea of the influence that criteria has on our alternatives.

Notice, in the process of doing AHP tree sensitivity we may only choose the criteria we wish to analyze, and we are then able to see the impact of that criteria on the alternatives. We would like to be able to do a similar analysis in the ANP case.

Consider, for example, that conventional AHP tree sensitivity can be applied to ANP, which is what is done in SuperDecisions. This is sometimes referred to as local priority sensitivity. However, there are additional connections in an ANP. You can talk about how important a node is with respect to another node. This works well for AHP trees because every node only has one parent, so that the connection is to the parent. In ANP, in contrast, there can be multiple connections (or parents) to one given node (a “fixed” node). In ANP, you can inquire how important a node is “with respect to” another node, since the connections in ANP are not automatically parent-child direction connections. Furthermore, the mathematics shows that no one node is “important” in an ANP network since one little change in one connection gets overwhelmed by all of the other data, due to all of the many connections.

In another conventional idea, discussed further below, the priority of a node (in the ANP) is changed after the limit matrix is calculated. That is, the node it is looked at after the fact. However, all of the ANP structure is ignored.

The technique referred to as global sensitivity tells you how important a node is, however, it is after all of the ANP limit matrix calculations have happened, so it essentially discards a lot of ANP information. It does not accurately tell how sensitive things are.

Sensitivity analysis, as it is conventionally used, is a very qualitative field. A user does not know what the quantitative difference is after making the change to a node. In practice, a user does a sensitivity analysis with the bar chart (as enabled by Decision Lens) to see how important the nodes are, such as by dragging a node all the way out to see that it has no influence.

1.2 Prior Existing ANP Sensitivity Ideas

We have already briefly mentioned most of the prior existing ANP sensitivity ideas. However we would like to collect them together here, and explain why we consider them to be insufficient analogues of AHP tree sensitivity.

Pairwise Comparison Sensitivity.

In this known analysis, a particular entry in a pairwise comparison matrix (and its reciprocal on the other side of the matrix) is changed, new local priorities are calculated, and the alternatives are re-synthesized accordingly. In order to do this a “with respect to” node is chosen, and two other nodes are chosen. Simply by virtue of all of these choices, this is not a sufficient analogue of AHP tree sensitivity. In addition, nothing useful is found in such analysis, since one pairwise comparison essentially never has an impact (except in a few degenerate cases).

Local Priority Sensitivity.

This known sensitivity technique amounts to changing a single entry in the unscaled supermatrix, recalculating the limit matrix, and re-synthesizing to arrive at alternative scores. In order to do this analysis we choose a “with respect to” node (the column of the supermatrix) as well as the row (the node whose priority we are changing). This method has two shortcomings. First we are not analyzing the sensitivity of a single node but rather of the node with respect to another node. Secondly, in nearly all cases, there is simply no sensitivity to witness (much as in the case of pairwise comparison sensitivity).

Global Priority Sensitivity.

In this known analysis, we tweak the global priority of a node (that is, after the limit matrix calculation has already occurred). This analysis proceeds by calculating the limit matrix, deriving global priorities from that limit matrix, then tweaking a node's global priority (and rescaling the others), and then re-synthesizing. This is problematic in several ways. First, if the model is a single level all calculations are done at the limit matrix level, and we are tweaking after that point so nothing useful has occurred. Second, even if the model is multiple level, by tweaking the global priority of a node after the limit matrix calculation our sensitivity analysis lies outside of much of the ANP theory, and thus feels somewhat foreign. It does have the advantage of showing the sensitivity of the model to a particular node, but at the cost of only working for multiple level models, and working outside of the context of the majority of ANP theory.

1.3 Proposed Solution

The present system is different, for reasons including that it can assign a value measuring how influential a node is. Consequently, one can identify the most influential node (or nodes). This metric might drive a user to reevaluate, e.g., their priorities (or pairwise comparisons) for that most influential node since priorities for that node makes a big difference to the ANP; or to spend more time evaluating the priorities of the more influential nodes. Alternatively, it might turn out that a small portion of nodes are most influential, and those nodes might be more heavily evaluated.

Consider that the ANP network models a decision, such as, a football team, a budget, a decision to buy a car, or other decisions which are usually complex and take into account various factors. The user can find out where in the analysis to focus their time by measuring sensitivity of different factors. For example, when the ANP network models a football team decision, the system and process helps the user decide whether to spend more time evaluating priorities with respect to the quarterback or the kicker? With a car, a user can determine whether to spend more time analyzing safety or price? As a post analysis step, a user can determine that, for example, of 30 nodes, only three are influential to the decision. By knowing that, the user can determine that, e.g., tolerance to risk affects the decision more than any other factor. In the past, one problem with conventional ANP is that where the numbers are coming from is a hidden process; this process and system can allow greater transparency to see where things are influencing the decision.

The problem we have is to get an ANP analogue of AHP tree sensitivity that yields similar results. The proposed solution can be summarized as taking the global priorities approach but moving it before the limit matrix calculation. Or, if one prefers, it can be summarized as simultaneously performing local sensitivity analysis on every column.

We want to obtain tree-sensitivity kinds of results from AHP into the ANP model context, with the same kind of usefulness. Further in accordance with exemplary embodiments, there is provided a system and processing in an ANP structure to get the same sorts of results.

Improved row sensitivity can be provided in an ANP network. The basic idea is to change every entry in the scaled supermatrix (and then rescale the rest).

The difficulty we face is determining how much to change each entry in the given row of the supermatrix by. In order to keep the analogy with AHP tree sensitivity, we would like to have a single parameter p that we vary between 0 and 1 (corresponding to the local weight in AHP tree sensitivity). By changing that single parameter we would be changing all of the entries in the given row of the scaled supermatrix (again we could do the same in the unscaled supermatrix, the difference in results is that one tells us how sensitive we are to the node globally as opposed to how sensitive we are to the node when viewed as a part of its parent cluster).

The question becomes, for each value of the parameter p, what should we change the entries in the given row of the scaled supermatrix to? There are many choices possible, however we will see that up to continuous change of the parameter there is only one choice which will preserve the “ANP structure” of the model. (This fuzzy terminology will be made precise in the coming pages. The basic idea of preserving “ANP structure” is that we do not change the node connections, and we leave ratios of local priorities as unchanged as possible.)

2 Supermatrix Row Perturbations which Preserve ANP Structure

The idea behind row sensitivity is to perturb (that is, change by a predetermined amount) each entry in a given row of the scaled supermatrix. In order to stay stochastic, when we perturb a single entry in the supermatrix we correspondingly change the rest of the entries in that column, so that the column still adds up to one. However, the “main change” in a column is to the entry in the given row, and the changes to the rest of the column could be seen as consequences of that original entry that is changed. Since we will be changing each entry in a row, we will be changing the rest of the entries so that the columns still add to one (by simply rescaling the rest of the entries in that matrix). In order to precisely describe what preserving ANP structure means, we use a bit of notation.

2.1 Notation and Definitions

We will use W for the weighted supermatrix, W_(i,j) for the entry in the i^(th) row j^(th) column of the weighted supermatrix. We have already mentioned that we want to use a single parameter p between 0 and 1 to describe the perturbation of our supermatrix. Let us define precisely what we mean now.

Part II Definition 1 (Entry perturbation). Let W be the weighted supermatrix of an ANP model. We say W′ is a perturbation of W in row i column j if:

-   -   W′ is stochastic of the same dimensions as W     -   The columns of W′ agree with the columns of W except for         possibly the j^(th) column.     -   The ratios of the entries in the j^(th) column of W′ are the         same as those of W except possibly the ratios involving the         i^(th) row.

Note 1. The above definition essentially says we have changed the entry in row i column j, and rescaled the remainder of the column so that the column still adds to one.

Part II Definition 2 (Matrix space). Let M_(r,k)(X) be the space of matrices with r rows, k columns, and entries in the space X.

Part II Definition 3 (Row perturbation). Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n). A family of perturbations of W in the r^(th) row is a continuous function f: [0, 1]→M_(n,n)([0, 1]) with the following properties.

1. f(p) is a stochastic matrix.

2. For some 0<p₀<1 we have f(p₀)=W. This p₀ is called the fixed point of the family.

3. f(p) is the result of a sequence of perturbations of W in row r column j as j ranges from 1 to n.

When the family of perturbations is clear, we will write W(p) for f(p), abusing notation in order to gain readability.

Part II Definition 4 (Trivial Column). Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n). Also fix a row 1≦r≦n to consider a family of row perturbations on. A column j of W is called a trivial column for row perturbations on row r (or simply a trivial column) if either the column is zero, or the column has all zero entries except the r^(th) entry is 1. A column is call non-trivial if it is not trivial.

2.2 Basic Properties Desired

There are two basic properties we would like a family of perturbations of the r^(th) row of the weighted supermatrix to have. They deal with the end points of the family as well as the general flow of the family. We will describe the properties as well as the reason for wanting those properties now.

Let's consider W(0). In considering the AHP tree analogy, the parameter p corresponds to the local weight of our node/criteria. So W(0) can reflect what happens when the r^(th) node is completely unimportant. In other words it can set all of the local weights for the r^(th) criteria to zero, i.e. make the r^(th) row of the supermatrix zero. The only question is what we can do with columns that have the r^(th) row's entry as a 1 (and thus the rest in that column are zero).

Trivial columns are unchanged for 0<p≦1 by construction. If trivial columns were to change at p=0 we would lose continuity at p=0. Thus, in order to preserve continuity we will keep trivial columns unchanged even when p=0. Next let's consider W(1). Again considering the AHP tree analogy, the parameter p being set to one places all importance on the node/criteria in question, and zeros out the rest. So the matrix W(1) can have the r^(th) row with l's in any column that had non-zero entries in W (the columns that had the r^(th) row with a zero means there was no connection there, so we should not change those values), and the rest remain zero.

Lastly in the AHP tree case, as the parameter increases the local priority (and hence global priority) increases. Because of the nature of feedback within an ANP model we cannot guarantee this global priority behavior. However we would like to have, as p increases the local priorities for the r^(th) criteria to increase (i.e. the values in that row of the weighted supermatrix). In other words the coordinate functions for the r^(th) row of the family of matrices W(p) are increasing functions.

2.3 Maintaining Proportionality

We reach a consideration about how a family of perturbations of the weighted supermatrix in a given row should behave. There are, of course, many ways we could perturb the values in a given row, based on the information of a single parameter (we could, for instance set all of the entries in that row to that parameter value). However not many of these choices would preserve the overall ANP structure, and this is what we consider now.

The idea is to maintain proportionality of elements in the supermatrix throughout our family as much as possible. We cannot keep all of the proportions identical since that would mean the matrix would never change (since the matrix needs to remain stochastic). In fact motivation comes from looking at the row we are perturbing and our axioms that W(0) can zero out that row and W(1) can place all importance on that row.

If we want to mimic AHP sensitivity, W(0) zeros out that row. By continuity this means that as p→0 W(p) should go to W(0). Thus, however we change that row we can make sure that as p→0 that row goes to zero. If we force ourselves to maintain proportionality in that row no matter what value p has (at least for p close to zero) we can achieve the desired result. For instance think of p as a scaling factor to multiply the row by. Then as p→0 that row does go to zero, and maintains proportionality. So it seems we can hope to have proportionality maintained in the row in question for small values of p.

However, considering W(1) shows this is not possible for values of p close to 1. For, if we maintain proportionality in that row, that row cannot go to 1 (in fact the best it could do is have one entry go to one, and the rest would maintain their proportionality to that one). Since it is not possible to maintain proportionality in that row and have that row go to one, we can look elsewhere for a position to maintain proportionality in. If we force the other rows to maintain their proportionality when p is close to 1, it turns out to maintain the proportionality of the distance from 1 of the entries in our row (which is a useful proportionality to maintain).

Thus the proportionality we expect to maintain depends on the values of our parameter p. Although no formal proof has been yet given that these proportionalities are possible we hope to have shown at least why we cannot have proportionality in the row in question as p goes to 1.

2.4 Formal Definition

We will now collect the various ideas presented above into a single definition for the kind of object we wish to study and use to extend the concept of AHP tree sensitivity to the ANP world.

Part II Definition 5 (Family of row perturbations preserving ANP structure). Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n). A family of perturbations of W in the r^(th) row f: [0, 1]→M_(n,n)([0, 1]) is defined as preserving the ANP structure if the following 1-5 are true:

1. Trivial columns (if present) remain unchanged throughout the family. In other words if the j^(th) column of W is trivial then the j^(th) column of f(p) equals the j^(th) column of W for all 0≦p≦1.

2. If W_(r,i) is zero then the i^(th) column of f(p)=W(p) equals the i^(th) column of W for all p (that is, if there is no connection from i to r we will not create one ever in the family).

3. If W_(r,i) is non-zero and the i^(th) column of W is non-trivial, then W(p)_(r,i) is not zero except for p=0 (that is the connection from i to r is not broken except when p=0 and all influence is removed from node r).

4. If p₀ is the parameter for which W(p₀)=W then for p<p₀W(p)'s r^(th) has the same proportionality as W's r^(th) row. That is, for p<p₀ we have W_(r,i)

$\frac{W_{r,i}}{W_{r,j}} = \frac{{W(p)}_{r,i}}{{W(p)}_{r,j}}$ where these fractions are defined.

5. For p>p₀ we have for all i, i′≠r:

$\frac{W_{i,k}}{W_{i^{\prime},j^{\prime}}} = \frac{{W(p)}_{i,j}}{{W(p)}_{i^{\prime},j^{\prime}}}$ where these fractions are defined. That is, maintain proportionality of all of the rows except for the r^(th) row.

6. We say that the family is increasing if W(p)_(r,i) is an increasing function if W_(r,i) is not zero, and is the constant function zero if W_(r,i)=0.

With this we have a definition of a family of row perturbations that preserve ANP structure, and good reasons to accept this as useful definition. However we do not yet know if such families exist.

2.5 Existence

In fact such families do exist, as we shall now prove. First we define our proposed family, and then prove it preserves the ANP structure.

Part II Definition 6. Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n), and fix r an integer between 1 and n. Pick 0<p₀<1, and define F_(W,r,p) ₀ : [0; 1]→M_(n,n)([0, 1]) in the following fashion. Firstly leave trivial columns unchanged throughout the family. Next, if 0≦p≦p₀ define F_(W,r,p) ₀ (p) by scaling the r^(th) row by

$\frac{p}{p_{0}},$ and renormalizing the columns. Since we have changed the entry in the r^(th) row and do not want to change the entry in the r^(th) row again by renormalizing, we instead scale the rest of the entries in that column to renormalize the columns. If p₀≦p≦1 define F_(W,r,p) ₀ (p) by leaving alone columns of W for which W_(r,i)=0 and scaling all entries in the other columns, except for the entry in the r^(th) row, by

$\frac{1 - p}{1 - p_{0}}$ and change the r^(th) entry to keep the matrix stochastic.

Note 2. There is a subtlety, in that we have defined the above function in two ways for p=p₀. However using either formula we get the result of W when we plug in p=p₀ so that the above function is well defined.

Note 3. The above function is a piecewise defined function whose pieces are linear, and they agree at the intersection of the two regions of definition. Thus the above function is continuous.

Theorem 1. Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n), fix r an integer between 1 and n, and pick 0<p₀<1. Then F_(W,r,p) ₀ (p) is a family of row perturbations preserving the ANP structure.

Proof. It is clear that F_(W,r,p) ₀ (p) satisfies the three conditions for being a family of row perturbations, thus we can proceed to demonstrating that it preserves the ANP structure. However the preservation of ANP structure simply follows from the definitions. In addition it is clear that F_(W,r,p) ₀ (p) is increasing as well.

2.6 Uniqueness

Thus we have a family of row perturbations which preserves the ANP structure, which is useful. However, what is surprising is that this family is essentially the only family preserving the ANP structure, up to change of parameter. Let us make this precise.

Theorem 2 (Uniqueness). Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n), fix r an integer between 1 and n, and pick 0<p₀<1. Let f(p) be a family of row perturbations preserving the ANP structure with p₀ as the fixed point. Then there exists a continuous map h: [0, 1]→[0, 1] so that f=F _(W,r,p) ₀ ∘h

Proof. We will define h(p) piecewise, first for 0≦p≦p₀ and then p₀≦p≦1. Let 0≦p≦p₀. Then f(p) preserves ratios in the r^(th) row, i.e. the r^(th) row is a scalar multiple of the r^(th) row of W, let j be a column for which W_(r,j)≠0. We can calculate that scalar as

$\frac{{f(p)}_{r,j}}{W_{r,j}}$ and thus we define

${h(p)} = {p_{0} \cdot {\frac{{f(p)}_{r,j}}{W_{r,j}}.}}$

Since f is continuous its (r, j) entry function is continuous and thus h is continuous. Notice that h(p₀)=p₀, and that we can determine F_(W,r,p) ₀ ∘h(p)_(r,j) using the following sequence of equalities.

$\begin{matrix} {{F_{W,r,p_{0}}\bullet\;{h(p)}_{r,j}} = {F_{W,r,p_{0}}\left( {h(p)} \right)}_{r,j}} \\ {= {F_{W,r,p_{0}}\left( {p_{0} \cdot \frac{{f(p)}_{r,j}}{W_{r,j}}} \right)}} \\ {= {W_{r,j} \cdot \left( {p_{0} \cdot \frac{{f(p)}_{r,j}}{W_{r,j}}} \right) \cdot \frac{1}{p_{0}}}} \\ {= {f(p)}_{r,j}} \end{matrix}$

Since f and F_(W,r,p) ₀ preserve the ANP structure and agree in the (r, j) entry, they agree in all entries. Thus for 0≦p≦p₀ f(p)=^(F) _(W,r,p) ₀ ∘h(p)

Next for p₀≦p≦1 we note that f(p) preserves the ratios of the rows other than r, since f preserves the ANP structure. Let W_(i,j) be a non-zero entry with i≠r. Since f preserves the ratios of rows other than the r^(th) row, we have a simple scalar multiplication of those rows. We can calculate that scalar as

$\frac{{f(p)}_{i,j}}{W_{i,j}}$ and we define h(p) for p₀≦p≦1 as

${h(p)} = {1 - {\frac{{f(p)}_{i,j}}{W_{i,j}}{\left( {1 - p_{0}} \right).}}}$

Notice that h(p) as defined above is continuous since f's entries are continuous and that h(p₀)=p₀ (thus both definitions agree at their overlap of p₀, so there is no ambiguity in our definition). Furthermore we can see the following equalities.

$\begin{matrix} {{F_{W,r,p_{0}}\bullet\mspace{11mu}{h(p)}_{i,j}} = {F_{W,r,p_{0}}\left( {h(p)} \right)}_{i,j}} \\ {= {\frac{1 - {h(p)}}{1 - p_{0}}W_{i,j}}} \\ {= {\frac{1 - \left( {1 - {\frac{{f(p)}_{i,j}}{W_{i,j}}\left( {1 - p_{0}} \right)}} \right)}{1 - p_{0}}W_{i,j}}} \\ {\frac{\frac{{f(p)}_{i,j}}{W_{i,j}}\left( {1 - p_{0}} \right)}{1 - p_{0}}W_{i,j}} \\ {= {f(p)}_{i,j}} \end{matrix}$

Since f and F_(W,r,p) ₀ preserve the ANP structure and agree in the (i, j) entry, they agree in all entries. Thus for p₀≦p≦1 f(p)=F _(W,r,p) ₀ ∘h(p)

Thus we have demonstrated h: [0, 1]→[0, 1] which is continuous (since the piecewise parts are continuous and they agree on the overlap) which satisfies f(p)=F _(W,r,p) ₀ ∘h(p) for all 0<p<1.

Remark 1. The previous theorem states that there is only one way to do row sensitivity in way that preserves the ANP structure (up to change of parameter).

3 Example Calculations

So that we may see how these results play out, let us consider a few examples calculated by hand.

3.1 Two node model

This model contains just two nodes in a single cluster, fully connected. The weighted supermatrix (which is really just the unweighted supermatrix in this case) is

$W = \begin{bmatrix} {.2} & \frac{2}{3} \\ {.8} & \frac{1}{3} \end{bmatrix}$

With this supermatrix we get the normalized priority vector for the alternatives (which we denote as A)

$A = \begin{bmatrix} {0.\overset{\_}{45}} \\ {0.\overset{\_}{54}} \end{bmatrix}$

We will do row sensitivity on the second row, using parameter values of 0.1 and 0.9 (which corresponds to pushing down the priority of the second row for p=0.1 and pushing it up for p=0.9). For simplicity we will use p₀=0.5.

As a matter of notation we will use A_(p) to denote the new synthesized normalized values of the alternatives when we do row sensitivity with value p, and L_(p) for the limit matrix when the parameter is p.

p=0.1: Let us calculate F_(W,2,0.5)(0.1) first (and then we will calculate the limit matrix). Using our formula we will scale row 2 by 0.1/0.5=0.2. Thus row two of our new matrix will be 0.4 and 0.2/3. Normalizing our columns we get the first row is 0.6 and 2.8/3. Thus

${F_{W,2,{.5}}(0.1)} = {\begin{bmatrix} {.6} & {2.8/3} \\ {.4} & {{.2}/3} \end{bmatrix}.}$

The limit matrix is therefore:

$L_{0.1} = \begin{bmatrix} {.7} & {.7} \\ {.3} & {.3} \end{bmatrix}$ which gives the new synthesized priorities of

$A_{0.1} = \begin{bmatrix} {.7} \\ {.3} \end{bmatrix}$ which has substantially reduced the score of the second alternative from the original values. This is what we would expect by analogy with AHP tree sensitivity. We have decreased the importance of the second alternative prior to calculating the limit matrix, and thus its overall priority has decreased after calculating the limit matrix.

p=0.9: Again let us calculate F_(W,2,0.5)(0.9) first and then proceed to the limit matrix. Using the definition we will scale the rows other than 2 (i.e. row one) by

$\frac{1 - 0.9}{1 - 0.5} = {0.2.}$ Thus the first row becomes 0.04 and 0.4/3. Renormalizing the columns yields the second row as 0.96 and 2.6/3. Thus

${F_{W,2,0.5}(0.9)} = \begin{bmatrix} {.04} & {{.4}/3} \\ {.96} & {2.6/3} \end{bmatrix}$

The limit matrix is therefore:

$L_{0.9} = \begin{bmatrix} {.121951219} & 0402955 & {.121951219} & 5777367 \\ {.878048780} & 9597046 & {.878048780} & 4222634 \end{bmatrix}$ which gives the new synthesized priorities of:

$A_{0.9} = \begin{bmatrix} {.121951} \\ {.878049} \end{bmatrix}$ which has substantially increased the score of the second alternative from the original values. Again this result is as we would expect.

3.2 Four Node Model

This is a model with two clusters each of which have two nodes (thus four nodes altogether). There is a single criteria cluster, and the alternatives clusters. In the criteria cluster there are criteria A and B. In the alternatives cluster are two nodes, alt1 and alt2. Everything in the model is fully connected and the weighted supermatrix, and alternative scores are as follows (the order of the nodes being A, B, alt1, and finally alt2).

$W = \begin{bmatrix} 0.375 & 0.20 & 0.175 & 0.10 \\ 0.125 & 0.30 & 0.325 & 0.40 \\ {{.0}{.400}} & 0.05 & 0.275 & 0.15 \\ 0.100 & 0.45 & 0.225 & 0.35 \end{bmatrix}$ $A = \begin{bmatrix} 0.388144 \\ 0.611856 \end{bmatrix}$

As before we will set p=0.1 first, then p=0.9, and we will work with criteria B sensitivity (i.e. row 2) and p₀=0.5.

p=0.1: First we calculate the new matrix. For p=0.1 we scale row 2 by 0.1/0.5=0.2, and then renormalize. We get

${F_{W,2,0,5}(0.1)} = \begin{bmatrix} {.417857} & {.268571} & {.242407} & {.153333} \\ 0.025 & 0.06 & 0.065 & 0.08 \\ {.445714} & {.067143} & {.380926} & {.536667} \\ {.111429} & {.604286} & {.311667} & {.349993} \end{bmatrix}$

The limit matrix result is:

$L_{0.1} = \begin{bmatrix} {.2572} & {.2572} & {.2572} & {.2572} \\ {.0598} & {.0598} & {.0598} & {.0598} \\ {.3248} & {.3248} & {.3248} & {.3248} \\ {.3583} & {.3583} & {.3583} & {.3583} \end{bmatrix}$

This yields the following synthesized priorities for alt1 and alt2.

$A_{0.1} = \begin{bmatrix} {.4758} \\ {.5242} \end{bmatrix}$

p=0.9: Let us calculate the new matrix. Using our formula we will multiply rows 1, 3, and 4 by

${\frac{1 - 0.9}{1.0{.5}} = 0.2},$ and then change row 2 to normalize the columns. This gives us

${F_{W,2,0.5}(0.9)} = \begin{bmatrix} {.075} & {.040} & {.035} & {.020} \\ {.825} & {.860} & {.875} & {.880} \\ {.080} & {.010} & {.055} & {.030} \\ {.020} & {.090} & {.045} & {.070} \end{bmatrix}$ the limit matrix is thus

$L_{0.9} = \begin{bmatrix} {.039761} & {.039761} & {.039761} & {.039761} \\ {.863877} & {.863877} & {.863877} & {.863877} \\ {.015210} & {.015210} & {.015210} & {.015210} \\ {.085178} & {.085178} & {.085178} & {.085178} \end{bmatrix}$ and finally the synthesized priorities are

$A_{0.9} = \begin{bmatrix} {.1515} \\ {.8485} \end{bmatrix}$

4 Alternate Definition of F_(W,r,p) ₀ (p)

The definition given previously for the family of row perturbations F_(W,r,p) ₀ (p) is useful conceptually; however, there is another useful way of defining that family (a different way to write the formula) that only talks about changing the r^(th) row and rescaling the rest of each column. We describe that formula in terms of the theorem below (stating that the new formulation is the same as our original formulation).

Theorem 3. Fix an ANP model (a single level of it) and let W be its weighted supermatrix (whose dimensions are n×n), and fix r an integer between 1 and n. Pick 0<p₀<1. We can define F_(W,r,p) ₀ : [0, 1]→M_(n,n)([0, 1]) in the following alternate fashion. First leave trivial columns unchanged throughout the family. Next, for all 0≦p≦1 we define F_(W,r,p) ₀ (p) by changing the r^(th) row and then rescaling the remaining entries in the columns so that the columns continue to add to one. For 0≦p≦p₀ we change the r^(th) by scaling it by

$\frac{p}{p_{0}}.$ For p₀≦p≦1 we change the entries in the r^(th) row by the following formula F _(W,r,p) ₀ (p)_(r,j)=1−α(1−W _(r,j)) where

$\alpha = {\frac{1 - p}{1 - p_{0}}.}$

Note 4. The above formulation implies that, for p₀≦p≦1 we scale the distance from 1 of the entries in the r^(th) row by

$\alpha = {\frac{1 - p}{1 - p_{0}}.}$

Proof. Our new definition agrees with the original definition for 0≦p≦p₀, thus we can proceed to the other case. Thus let p₀≦p≦1. We have the formula F _(W,r,p) ₀ (p)_(r,j)=1−α(1−W _(r,j)).

Fix a non-trivial column j, we can show that

${F_{W,r,p_{0}}(p)}_{i,j} = {{\frac{1 - p}{1 - p_{0}}W_{i,j}} = {\alpha\; W_{i,j}}}$ for all i≠r to prove our definitions coincide.

Let β_(j) be the scaling factor we scale the entries of the j^(th) column by (except for the r^(th) row).Then F_(W,r,p) ₀ (p)_(i,j)=βW_(i,j). Since the j^(th) column of F_(W,r,p) ₀ (p) adds to one, we get the following sequence of equalities.

$\begin{matrix} {1 = {\sum\limits_{i = 0}^{n}{F_{W,r,p_{0}}(p)}_{i,j}}} \\ {{= {{F_{W,r,p_{0}}(p)}r}},{j + {\sum\limits_{i \neq r}{F_{W,r,p_{0}}(p)}_{i,j}}}} \\ {= {1 - {\alpha\left( {1 - W_{r,j}} \right)} + {\sum\limits_{i \neq r}{\beta_{j}W_{i,j}}}}} \\ {= {1 - {\alpha\left( {1 - W_{r,j}} \right)} + {\beta_{j}{\sum\limits_{i \neq r}W_{i,j}}}}} \\ {= {1 - {\alpha\left( {1 - W_{r,j}} \right)} + {\beta_{j}\left( {1 - W_{r,j}} \right)}}} \end{matrix}$

The last equality coming from the factor that the columns of W add to one. We can continue in the following fashion.

1 = 1 − α(1 − W_(r, j)) + β_(j)(1 − W_(r, j)) α(1 − W_(r, j)) = β_(j)(1 − W_(r, j)) α = β_(j)

Thus we are rescaling the entries of the j^(th) column (except the entry in the r^(th) row) by α, which completes the proof

In review, there are two different definitions of the above approaches, Part II, Section 2.5 and Part II, Section 4, which is an alternate. Part II, Section 4 can be easier to code as software, but it is equivalent to the definition of Part II, Section 2.5.

A difference between the definitions of Part II, Section 2.5 and Part II, Section 4 is in how the r^(th) row is changed. For 0≦p≦p₀, is about perturbing downward (scaling by p/p₀). Perturbing downward is identical in both definitions.

In Part II Definition 6, for perturbing downward (0≦p≦p₀) the given row is rescaled; for perturbing upward (p₀≦p≦1), everything except the given row is rescaled by a particular factor. Mathematically, this is straightforward. Calculationally, it is difficult.

From a calculational perspective, it is easier work with one row. In Part II, Section 4, perturbing downwardly is the same as Part II, Section 2.5 (rescale the given row by p/p0). For perturbing upwards, we rescale the given r^(th) row by the given formula in Part II, Section 4, Theorem 3. That is, whether we perturb upward or downward, we change the r^(th) row, and then we rescaled the remaining rows. If perturbing downward, rescale by p/p₀. If perturbing upward, change the entries by the given formula in Part II, Section 4, Theorem 3 (which is rescaling to keep the distances from 1 the same). Part II, Section 2.5 performs the upward perturbation differently, as discussed above.

The reason the upward and downward perturbation approaches are different is due to end point behavior. As a node is perturbed upward, the priorities approach 1 (which adds importance on that node). As the priorities for a node are perturbed downward to approach zero, less importance is placed on that node. The same formula will not provide behavior for upward and downward. As approaching 0, the node gets less important and priorities approach 0. As importance approaches 1, all other nodes get more inconsequential.

The approach of Part II, Sections 2.5 and 4 will now be discussed in a more general sense. Referring now to FIG. 6, a diagram illustrating a measurement of sensitivity of a node in an ANP weighted supermatrix will be discussed and described. At (1) is a starting ANP weighted supermatrix 601, which has been prepared in accordance with conventional techniques resulting in the illustrated entries for each local priority. That is the value of N1 with respect to N1 is 0.1, N2 with respect to N1 is 0.3, N3 with respect to N1 is 0.6, N1 with respect to N2 is 0.2, N2 with respect to N2 is 0.6, N3 with respect to N2 is 0.2, N1 with respect to N3 is 0.4, N2 with respect to N3 is 0.1, and N3 with respect to N3 is 0.5.

At (2), the sensitivity of a node is transformed. That is, a node (sometimes referred to as a “fixed node”) is selected and the priorities of the selected node are perturbed. In the illustration, the selected node, N2, corresponds to the middle row and the priorities are perturbed upward. In this example, the predetermined fixed point Po and parameter value P selected for use in the sensitivity transformation are 0.5 and 0.75, respectively.

At (3) is an ANP weighted supermatrix 603 which has sensitivity of a row corresponding to the selected node perturbed upwardly. To arrive at the row sensitivity perturbed ANP weighted supermatrix 603, the proportionality of the starting ANP weighted supermatrix 601 has been maintained despite perturbing the selected node N2, and the proportionality is substantially present in the row sensitivity perturbed ANP supermatrix 603, with the exception of the selected node which was perturbed. As summarized in this illustration, the values in the middle row (corresponding to the selected node which is perturbed) of the supermatrix are made larger, whereas the values in the other rows are made smaller.

Since p₀ is 0.5 and p is 0.75, p is moving half way to 1. Proportionally, then, the value at N2, N2 should move halfway to 1. The value at N2, N2 is 0.6, which is 0.4 from 1. By adding 0.2 to 0.6 (i.e., 0.8), then N2, N2 will be perturbed halfway to 1. The generation of the row sensitivity perturbed matrix continues as detailed above.

At (4) the sensitivity of the node which was perturbed is measured (also referred to as “assessed”). The assessment can include determining the sensitivity of the selected node before and after perturbation. Sensitivity is defined to be the new synthesized alternatives priority. Sensitivity is a value x, 0<x<1. By perturbing one or more selected nodes according to a predetermined amount, the sensitivity of the selected node with respect to the ANP model can be quantified.

Referring now to FIG. 7, an explanatory diagram for a further explanation of FIG. 6 will be discussed and described. FIG. 7 is a visualization of the relation of the three nodes N1, N2, and N3 701, 705, 703. The directional “pipes” from one node to another which reflect the importance. As in FIG. 6, here the sensitivity of node N2 is being measured and hence the size of pipes that end in node N2 will be increased, i.e., pipes from N1 to N2, N2 to N2, and N3 to N2. The sizes of the other pipes are decreased, in proportion to the increase.

The same proportionality in the ANP weighted supermatrix can be maintained while preserving the ANP structure. The proportionality is maintained throughout the change of priorities of the node in the ANP weighted supermatrix to be less important and/or more important, as well as throughout the assessment of the sensitivity of the node which was changed relative to the ANP model.

To preserve the ANP structure, connections are not created or destroyed. That is, an entry in the matrix is not changed to or from zero, except when p=0, since a non-zero value represents a connection whereas a zero value indicates that there is no connection.

Preserving proportionality is a more difficult consideration. So, the question is, if I am making changes to this row, what is the most proportionality I can keep? Hence, maintaining proportionality is the more difficult and/or subtle problem for figuring out how this should behave.

No connections in the ANP network are created or destroyed by doing this present process. If a priority is zero, then there is no connection to another node. If that is ever changed from zero to something, then a connection has been created by the system, which is bad because the user did not create the connection. The change from zero changes the ANP structure because it creates a connection that was not there originally. Likewise, taking a non-zero value (which is a connection) and changing it to zero deletes a connection which was there. An embodiment of the present process does not create or destroy connection.

Now consider how to preserve as much of proportionality as possible, that is, preserving the ratios of the numbers involved in the ANP model as possible. If, in the original ANP model, e.g., node A is twice as good as node B, that proportionality is maintained as much as possible. It cannot be kept exactly, because that means nothing can be changed. However, by doing row sensitivity, you will break a few proportionalities. There is no choice. But, the other proportionalities you want to keep.

That is, to keep proportionality while changing a node to test that node, you are attempting to maintain proportionality for the other non-changed nodes, as well as that row as much as possible. Proportionality involves a node and a with-respect-to, and you want to preserve those proportionalities as much as possible. Part II, Section 2.3 (above) further discusses maintaining proportionality.

To measure sensitivity, a row will be changed. There is one way to change that row to keep as much proportionality throughout the ANP network as possible. Preserving ANP proportionality is discussed for example in Part II, Section 2.4, and Part II Definition 5.

While maintaining proportionality, trivial columns are not changed. This is discussed above, for example, in Part II, Section 2.4, point 1, and “trivial columns” are defined in Part II Definition 4. That is, something that is not from, stays that way; or something that is only connected to the fixed node, stays that way.

While maintaining proportionality, connections are not created, as discussed in, e.g., Part II, Section 2.4, point 2. Also, as discussed in Part II, Section 2.4, point 3, connections are not destroyed. To summarize points 2 and 3, in order to preserve ANP structure, connections are not created or destroyed.

Preservation of proportionality is further discussed in Part II, Section 2.4, points 4 and 5. There are two cases discussed. There is the case of perturbing downward, and the case of perturbing upward. Case 4 (“perturbing downward”) is decreasing the influence/importance of a node to look at its sensitivity. Case 5 is increasing the importance of a node to look at its sensitivity. Both cases are going to tell you what kind of proportionality is to be maintained.

The concept of row sensitivity opens up many avenues of analysis not previously available in ANP theory. For instance, there is influence analysis, i.e. which node is most influential to the decision the ANP model is making. Another example would be perspective analysis, which tells how important the alternatives would be if a single node was the only one in the model with weight (however we do not forget the rest of the model in this calculation). Yet another example is marginal analysis, that is, what are the rates of influence of each of the nodes (a derivative calculation). A final example applying row sensitivity would be search for highest rank influence (that is, which node causes rank change first).

Part III: ANP Influence Analysis

ANP Influence Analysis

A fundamental question for ANP models is which nodes are the most or least influential to the decision the model represents. ANP row sensitivity opens up many avenues of attack on this problem. In the following Part, we present here one such attack, which involves using the row sensitivity calculation combined with different “metrics” (these distance measures are not metrics in the topological sense, rather they loosely calculate distances). Hence, a new terminology is used to refer to these things, namely, “metriques.”

1 Introduction

After an ANP model is created and yields synthesized values for the alternatives we would like to understand how the structure and numerics of the model affect the results of the model. In traditional AHP tree models we can use sensitivity to increase or decrease the importance of a given node, and see how the alternatives change. With the advent of ANP row sensitivity we can perform a similar analysis on ANP models. However, this only yields a weak qualitative analysis of the situation (that is we can only roughly tell that this node appears to move alternatives more or less than the others). A more desirable analysis would be a single non-negative numerical value that describes the quantity of influence for each node.

1.1 Concept of Influence Analysis

The fundamental concept behind ANP influence analysis outlined here is that we wish to combine ANP row sensitivity (the ANP analogue of tree sensitivity) with distance measures describing how far alternative values move in the process of sensitivity. There are two subtleties to handle in doing this analysis. The first is how to use ANP row sensitivity to move the alternatives, and the second is how we will measure distances traveled. We shall deal with the latter first, and the former in the following section.

1.2 Distance Measures and Metriques

There is a whole branch of mathematics devoted to studying distance measures in spaces (metric spaces). Unfortunately, the kind of distance measures we will use sometimes fall outside of this theory (for instance percent change distance). As a result we need to be a bit careful in our terminology from the outset. We cannot call these things “metrics” since those objects have a precise mathematical definition to which we do not wish to limit ourselves. Instead we will use the terminology of a metrique to describe the distance measures we will be using.

Part III Definition 1 (Metrique). Let X be a space, d:XxX→Ra continuous function is a metrique iff for all x, yεX.

1. d(x, y)≧0

2. d(x, x)=0

Note 1. For those with some knowledge of metrics notice there is neither triangle inequality, nor symmetry, nor even an assertion that d(x, y)=0 iff x=y. This is a very weak cousin of traditional metrics.

1.3 Review of ANP Row Sensitivity

The following is a brief review of the concepts involved in ANP Row Sensitivity. The purpose of ANP row sensitivity is to change all of the numerical information for a given node in a way that is consistent with the ANP structure, and recalculate the alternative values (much as tree sensitivity works). We do this by having a single parameter p that is between zero and one, which represents the importance of the given node. There is a parameter value p₀ (called the fixed point) which represents returning the node values to the original weights. For parameter values larger than p₀ the importance of the node goes up, and for parameter values less than p₀ the importance of the node goes down. Once the parameter is set, this updates values in the weighted supermatrix (although it can also be done with the unscaled supermatrix, working by clusters instead) and re-synthesizes. There is essentially one way to do this calculation and preserve the ANP structure of the model. In the notation of that paper, let W be the weighted supermatrix of a single level of our model, ANP row sensitivity constructs a family of row perturbations of W. A family of row perturbations of W is a mapping f: [0,1]→M_(n,n)([0,1]) that gives a weighted supermatrix f(p) for each parameter value pε[0,1]. This mapping must preserve the ANP structure of our original supermatrix. The only real choice is what to make our fixed point p₀. Once we have chosen that, the standard formula for the family of row perturbations of row r of W preserving the ANP structure is labeled F_(W,r,p) ₀ : [0,1]→M_(n,n)([0,1]) and is defined in the following way.

1. Leave trivial columns unchanged. A trivial column is either a zero column, or a column with all zeroes except one entry that is one.

2. If 0≦p≦p₀ define F_(W,r,p) ₀ (p) by scaling the r^(th) by

$\frac{p}{p_{0}}$ and scaling the other entries in the columns so as to keep the matrix stochastic.

3. If p₀≦p≦1 define F_(W,r,p) ₀ (p) by leaving alone columns of W for which W_(r,i)=0 and scaling all entries in the other columns, except for the entry in the r^(th) row, by

$\frac{1 - p}{1 - p_{0}}$ (and change the entry in that r^(th) row so as to keep the matrix stochastic).

2 Influence Analysis

The idea behind influence analysis (as mentioned before) is to use ANP row sensitivity on a given node, and then create a score based on how much the alternative scores change. There are many ways we can use ANP row sensitivity to attempt to understand influence, several of which will be outlined in subsequent papers. (It turns out that many of the ways one might try to use ANP row sensitivity for influence analysis really give other kinds of information than influence.) For now we focus on a particular method and explain why we are using that method.

Let us fix a node that we wish to analyze, let p be our parameter for ANP row sensitivity, and p₀ be the fixed point for the family as defined in ANP row sensitivity. There are two possible directions of influence, namely increasing p above p₀ and decreasing p below p₀. Thus any influence analysis must check both increasing and decreasing values of p. There are several natural possibilities for changing the parameter p which we outline below.

Infinitesimal. We can do a small change in p above p₀ and below p₀ and look at the rate of change based on that. In the limit this is a derivative calculation, which we call “marginal influence” analysis. Although useful, this only tells how much influence a node has nearby the current values (it may have large marginal influence, but incredibly small influence after moving 0.001 units of p, for instance). This is a standard problem of using a rate of change to measure something about the original quantity. Namely one only knows the instantaneous rate of change at a point, and that rate of change may change dramatically nearby (thus the quantity may not change much even if the rate of change is large, if the rate of change drops to zero quickly).

Component. We can calculate the limit as p goes to 1.0. This does tell us a form of influence. However, if we consider what that calculation means, it means we are taking nearly all of the priority from other nodes and giving them to our node. This essentially is telling us what the synthesis looks like from the perspective of the given node, and not directly telling us the influence of that node.

Our influence analysis. We could fix a parameter value larger than p₀, denoted p₊ and fix a parameter value smaller than p₀, denoted p⁻. We can then move the parameter p to those two values and consider how far the alternatives have moved. In that way we can compare the distance the alternatives are moved depending on which node we use. We will use the lower/upper bound method to determine influence, in part because the infinitesimal and component methods outlined above do not show influence, but other useful information. By moving the parameter p to the same lower and upper values for each node (which corresponds to changing the importance of each node by the same amount) we can see which node affects the synthesized values for the alternatives most. In order to compare which node influences the alternative scores most we need a metrique to describe how far the alternatives have traveled from their initial values.

2.1 Metriques Used

There are several metriques which are natural to use to compare one set of alternative scores to another set. The following are some standard metriques we used in analyzing the examples given at the end (not all of these are reported in the examples, only the ones we have found most useful for the given calculation). However, it is by no means to be considered all inclusive. All of the metriques below are on the space R^(n). Let x=(x ₁ ,x ₃ , . . . ,x _(n)) y=(y ₁ ,y ₂ , . . . ,y _(n)) be two vectors in R^(n) (we will use these to write the formulas for each of the following):

Taxi cab: This is the standard taxi cab metric. The taxi cab distance between x and y is given by taxi cab distance=|x ₁ −y ₁ |+|x ₂ −y ₂ |+ . . . +|x _(n) −y _(n)|

Percent change: This is the sum of percent changes in the components of x and y. Since we are allowing components of x to be zero, we need to be careful in defining percent change there. This case will happen very infrequently in actual ANP sensitivity. Since it is impossible to define percent change from a 0 starting value, we define it to be 0. The formula is given by

${{percent}{\mspace{11mu}\;}{change}\mspace{14mu}{distance}} = {\sum\limits_{i = 1}^{n}\left\{ \begin{matrix} {\frac{x_{i} - y_{i}}{x_{i}}} & {if} & {x_{i} \neq 0} \\ 0 & {if} & {x_{i} = 0} \end{matrix} \right.}$

Maximum percent change: This is similar to the previous metrique. The only difference being we pick the maximum percent change component instead of summing the components.

Rank change: This is a simple formulation of how much the ranking of vector x and y differ. The ranking of x simply is the information of which component of x is largest, second largest etc, and is stored in an integer vector r^(x)εZ^(n). Where r^(x) _(i) is the ranking of the i^(th) component of x. (For instance if x=(0.3, 0.1, 0.2, 0.7) then r^(x)=(2, 4, 3, 1) because the largest component of x is the fourth, thus r^(X) ₁=1. The second largest component of x is the first component, thus r^(x) ₁=2.) The ranking change metrique just takes the taxi cab distance between r^(x) and r^(y) (that is the difference of the rankings of each vector).

2.2 Combinations of Metriques

Each of the metriques mentioned above measures distances slightly differently. A small overall change can cause many rank changes (and likewise a large overall change can leave rankings unchanged). Similarly a 100% change of the value 0.01 only changes it to 0.02 which is a small overall change (even though it is a large percent change). Thus there is no clear choice about which metrique to use in all circumstances.

We can remedy this by making the choice on a per model basis. To give us flexibility we could take a weighted average of the different metriques. We could have a metrique that weighs rank changes highly, and percent changes next highest, and finally gives a small amount of weight to taxi cab changes. We can picture this as having a tree sensitivity view where the nodes are the different metriques available with scores next to them (these scores would always add to one). We could weight one metrique higher by dragging the bar next to that metrique out longer (and thereby shortening the remaining metrique's bars).

2.3 Lower and Upper Parameter Value

The lower parameter value p⁻ and upper value p₊ must be fixed for any particular influence analysis (although clearly we are free to choose different lower and upper values to compare with at a later point). That is, we must use the same values for p⁻ and p₊ for each node in the model when doing influence analysis. However, after that influence analysis is completed we may choose to use different values, and compare the results. Such a varied approach gives us useful information. There are several issues which can be addressed by varying these values.

By choosing values of p⁻ and p₊ close to, and equidistant from p₀ we can see how much influence the nodes have for smaller changes.

By choosing values of p⁻ and p₊ far from, and equidistant from p₀ we can see how much influence the nodes have for large changes.

We can also break the equidistant rules mentioned above. Although keeping the upper and lower value equidistant from the starting value may appear to be a good approach, it has one significant drawback. Namely lower parameter values have far less influence by their nature. That is, moving p to one places all priority on the given node and takes away every other nodes priority (a huge change). However, moving p towards zero moves priority away from the given node and proportionately redistributes that priority to the rest of the nodes (a much smaller change). We can remedy this inequality by pushing p⁻ further away from p₀ than p₊ is.

3 Examples

Throughout the examples we use the family of row perturbations defined in ANP row sensitivity. The following examples are calculated with p⁻=0.1 and p₊=0.9 unless otherwise marked. These are large values for the upper and lower bounds. However they do reveal both interesting and useful changes. Perhaps most surprisingly they reveal many nodes with little to no influence whatsoever.

3.1 4node2.mod

This model has two clusters, “A1 criteria” and “Alternatives”. There are two criteria “A” and “B”, and two alternatives “1” and “2”. All nodes are connected to each other, and the weighted supermatrix is as follows (the ordering of nodes in the supermatrix is “A”, “B”, “1”, “2”).

$W = \begin{bmatrix} 0.3750 & 0.2000 & 0.0500 & 0.3333 \\ 0.1250 & 0.3000 & 0.4500 & 0.1667 \\ 0.3333 & 0.0500 & 0.2750 & 0.1500 \\ 0.1667 & 0.4500 & 0.2250 & 0.3500 \end{bmatrix}$

Keeping with the notation of ANP row sensitivity setting the parameter to 0.1 in row 1 we get the following new scaled supermatrix.

${F_{W,1,0.5}\left( {0,1} \right)} = \begin{bmatrix} 0.0750 & 0.0400 & 0.0100 & 0.0667 \\ 0.1850 & 0.3600 & 0.4690 & 0.2333 \\ 0.4933 & 0.0600 & 0.2866 & 0.2100 \\ 0.2467 & 0.5400 & 0.2345 & 0.4900 \end{bmatrix}$

The limit matrix is:

$L = \begin{bmatrix} 0.047829 & 0.047829 & 0.047829 & 0.047829 \\ 0.315993 & 0.315993 & 0.315993 & 0.315993 \\ 0.190763 & 0.190763 & 0.190763 & 0.190763 \\ 0.445415 & 0.445415 & 0.445415 & 0.445415 \end{bmatrix}$ which gives us the following synthesized alternative scores shown in Part III, Table 1.

Part III, Table 1: Alternative Normal Ideal Raw 1 0.299858 0.428281 0.190763 2 0.700142 1.000000 0.445415

The following Part III, Table 2 is a collection of results setting upper and lower parameter values for each node and the corresponding changes in output as actually computed by a software implementation.

Part III, Table 2: Param Max % chg Rank chg Alt 1 Alt 2 Original 0.5 0.0000 0 0.39 0.61 A:high 0.9 0.5376 2 0.63 0.37 B:high 0.9 0.7324 0 0.15 0.85 1:high 0.9 0.9414 2 0.94 0.06 2:high 0.9 0.9425 0 0.04 0.96 A:low 0.1 0.3415 0 0.3 0.7 B:low 0.1 0.4238 0 0.48 0.52 1:low 0.1 0.8206 0 0.1 0.9 2:low 0.1 0.7550 2 0.8 0.2

Since the alternatives are also nodes we can view the influence of the alternatives on the decision. In rare instances this may be useful, however, most of the time this is a fairly useless calculation. So to analyze this, we really need only consider the criteria “A” and “B”. Part III, Table 3 of values for them alone is:

Part III, Table 3: Param Max % chg Rank chg Alt 1 Alt 2 Original 0.5 0.0000 0 0.39 0.61 A:high 0.9 0.5376 2 0.63 0.37 B:high 0.9 0.7324 0 0.15 0.85 A:low 0.1 0.3415 0 0.3 0.7 B:low 0.1 0.4238 0 0.48 0.52

Looking at this table we already see something interesting. By maximum percentage change node “B” appears to be the most influential. However node “A” is the one that gives rise to a rank change (changing upwards the influence of node “A”). Thus if we are scoring rank changes higher than movement of alternative scores, node “A” would be considered the most influential. If we are scoring movement of alternatives scores higher node “B” would be the most influential. If we allow weighting of these various metrics we can arrive at a blending of these results that would most match the preferences one has on the importance of the various metriques.

3.2 BigBurger.mod

The following is the table of influence analysis as generated by our software implementation, for the model BigBurger.mod that is included in the Super Decisions sample models directory. The results in Part III, Table 4 have been sorted on the maximum percent change column.

Part III, Table 4: Param Max % chg Taxi Cab McD BK Wen Original Values 0.5 0.00% 0 0.63 0.23 0.13 1 Subs 0.9 231.91% 0.87 0.41 0.31 0.28 5 Drive Thru 0.9 52.75% 0.28 0.77 0.13 0.1 1 White Collar 0.9 51.76% 0.2 0.73 0.2 0.07 3 Students 0.9 48.65% 0.17 0.71 0.21 0.08 2 Blue Collar 0.9 48.64% 0.17 0.71 0.21 0.08 2 Recycling 0.9 45.77% 0.22 0.73 0.18 0.08 4 Families 0.9 45.20% 0.12 0.69 0.23 0.08 3 Parking 0.9 39.45% 0.17 0.71 0.2 0.09

There are a few things to note here. First a rank change column was not included because none of these caused any rank changes. Second, only the top few scoring nodes were included in this list. Third, the node “1 Subs” is clearly the most influential by a large margin. Even though “1 Subs” does not cause a rank change, it does make a huge change in the numerics of the result.

Using value of p⁻=0.1 and p₊=0.9 are rather large changes. If we wish to view smaller changes we can use 0.3 and 0.7 respectively. Using those values, sorting and showing the top few results we get the following Part III, Table 5.

Part III, Table 5: Param Max % chg McD BK Wen Original Values 0.5 0.00% 0.63 0.23 0.13 1 Subs 0.7 80.62% 0.53 0.26 0.2 5 Drive Thru 0.7 27.95% 0.7 0.19 0.12 1 White Collar 0.7 25.92% 0.68 0.22 0.11 3 Students 0.7 23.13% 0.67 0.22 0.11 2 Blue Collar 0.7 23.08% 0.67 0.22 0.11 2 Recycling 0.7 23.05% 0.68 0.21 0.11 4 Families 0.7 22.72% 0.66 0.23 0.11 3 Parking 0.7 21.88% 0.68 0.21 0.11

Since we are making smaller changes in the parameter, the resulting maximum percentage change is smaller. However, we still get the same ordering of the top few scoring nodes.

Referring now to FIG. 8, a data flow diagram illustrating a measurement of change distance of nodes in an ANP weighted supermatrix will be discussed and described. FIG. 8 provides an overview of the various techniques discussed in greater detail in this Part. FIG. 8 illustrates an ANP matrix 801 generated using known techniques. Values are represented in the illustration by an “x”. The ANP model represents factors in a decision. In this example, there are three nodes N1, N2, N3, representative of two or more nodes in an ANP model. How to set up the ANP weighted supermatrix 801 so that it represents a decision and the factors involved are well known and the reader is assumed to be familiar with these basic principals in initially setting up a supermatrix.

After the ANP supermatrix 801 has been generated, one of the nodes is fixed 803, and row sensitivity of the node is measured using (i) a predetermined increase value, and/or (ii) a predetermined decrease value. For the entire duration that the node is fixed and the row sensitivity is measured, the same proportionality in the ANP weighted supermatrix is maintained, for all of the nodes. As a part of performing row sensitivity, synthesized alternative scores are changed.

Then, a distance change value is generated 805 for the node on which row sensitivity was performed, based on how much the synthesized alternative scores traveled during the ANP row sensitivity. One or more metrique calculations are provided, in order to compare the way that different nodes influence alternatives scores.

In this example, four metrique calculations are provided, and the distance change value for the node(s) can be run through one or more of the metrique calculations. In this example, the metrique calculations are a taxi cab metrique 807, a percent change metrique calculation 809, a maximum percent change metrique 811, and a rank change metrique 813. The taxi cab metrique 807 measures how far the alternatives score has been moved, that is, the distance, e.g., a change from 0.01 to 0.02 is 0.01. The percent change metrique calculation 809 measures how much change there was from the starting value, e.g., a change from 0.01 to 0.02 is a 100% change. The maximum percent change metrique 811 looks at the largest percent change in an alternative's scores. The rank change metrique 813 formulates how much the rankings were changed by the row sensitivity, e.g., when the largest component changed to become the fourth largest.

To provide a single score per node which reflects the distance change value and scores, the set of alternative scores from two or more metriques are combined 815 into a single score for the node. For example, the scores can be averaged, and the average can be weighted. The weighting can be selected depending on what is more significant. For example, if it is most significant when rankings are changed, the rank change metrique can be weighted more heavily in the average than other metriques. Techniques for combining scores and preparing averages are known. The combination step 815 can be skipped if not desired, for example, if there is only one metrique calculation or if separate values for each of the individual metriques are desired.

Then, it may be desirable to compare 817 a set of the alternative scores developed from the above-illustrated metriques calculations 807, 809, 811, 813 (or the combined metriques) to a set of alternative scores for another node. Conveniently, the alternative scores for the other node(s) can be tabulated, such as in illustrated table 819 of results.

In table 819 of results, the nodes are listed as well as the row sensitivity increase (e.g., N1HIGH, N2HIGH, N3HIGH) or decrease (e.g., N1LOW, N2LOW, N3LOW). The designations “TAXI”, “% CHG”, “MAX % CHG”, and “RANK CHG” are illustrated as representative of the calculated result values (illustrated for example in Part III, Tables 2, 3, 4 or 5).

This disclosure is intended to explain how to fashion and use various embodiments in accordance with the invention rather than to limit the true, intended, and fair scope and spirit thereof. The invention is defined solely by the appended claims, as they may be amended during the pendency of this application for patent, and all equivalents thereof. The foregoing description is not intended to be exhaustive or to limit the invention to the precise form disclosed. Modifications or variations are possible in light of the above teachings. The embodiment(s) was chosen and described to provide the best illustration of the principles of the invention and its practical application, and to enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims, as may be amended during the pendency of this application for patent, and all equivalents thereof, when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled. 

What is claimed is:
 1. An apparatus comprising: an analytic network process (ANP) storage memory that stores an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model; and a processor in communication with the ANP storage memory, the processor being configured to facilitate performing a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model.
 2. The apparatus of claim 1, wherein the processor is configured to facilitate determining a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model.
 3. The apparatus of claim 1, wherein the processor is configured to indicate the nodes determined to have the most influence.
 4. The apparatus of claim 1, wherein the processor is configured to prompt a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model.
 5. The apparatus of claim 1, wherein the processor is configured to provide, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model.
 6. The apparatus of claim 1, wherein the processor is configured to base the rank of each node in the ANP model on an amount of influence each said node has on rankings of alternatives in the ANP model.
 7. The apparatus of claim 1, wherein the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model.
 8. The apparatus of claim 1, further comprising an input unit configured to input, from an input device to the processor, pairwise comparisons, ANP ratings, or ANP client data, which are stored as the data in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values.
 9. A method, comprising: storing, in an analytic network process (ANP) storage memory, an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model; and in a processor in communication with the ANP storage memory, performing a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model.
 10. The method of claim 9, further comprising determining a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model.
 11. The method of claim 9, further comprising outputting, to an output unit in communication with the processor, an indication of the nodes determined to have the most influence.
 12. The method of claim 9, further comprising prompting a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model.
 13. The method of claim 9, further comprising providing, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model.
 14. The method of claim 9, wherein the rank of each node in the ANP model is based on an amount of influence each said node has on rankings of alternatives in the ANP model.
 15. The method of claim 9, wherein the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model.
 16. The method of claim 9, further comprising inputting, from an input device, pairwise comparisons, ANP ratings, or ANP client data, which are transformed into priority vectors and stored as the data stored in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values.
 17. A non-transitory computer-readable storage medium encoded with a computer executable instructions, wherein execution of said computer executable instructions by one or more processors causes a computer to perform the steps of: storing, in an analytic network process (ANP) storage memory, an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model; and in a processor in communication with the ANP storage memory, performing a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model.
 18. The non-transitory computer-readable storage medium of claim 17, further comprising determining a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model.
 19. The non-transitory computer-readable storage medium of claim 17, further comprising outputting an indication of the nodes determined to have the most influence.
 20. The non-transitory computer-readable storage medium of claim 17, further comprising prompting a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model.
 21. The non-transitory computer-readable storage medium of claim 17, further comprising providing, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model.
 22. The non-transitory computer-readable storage medium of claim 17, wherein the rank of each node in the ANP model is based on an amount of influence each said node has on rankings of alternatives in the ANP model.
 23. The non-transitory computer-readable storage medium of claim 17, wherein the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model.
 24. The non-transitory computer-readable medium of claim 17, further comprising instructions for: prompting a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model.
 25. The non-transitory computer-readable storage medium of claim 17, further comprising instructions for: inputting, from an input device, pairwise comparisons, ANP ratings, or ANP client data, which are transformed into priority vectors and stored as the data stored in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values.
 26. An apparatus comprising: an analytic network process (ANP) storage memory that stores an ANP model with feedback connections in place among nodes within the ANP model and with at least preliminary input of data stored in the ANP model; and a processor in communication with the ANP storage memory, the processor being configured to: perform a rank influence analysis on the ANP model, wherein the rank influence analysis determines one or more nodes among the nodes in the ANP model in which a smallest upward or downward change in importance will change a rank among the nodes in the ANP model, so as to determine which one or more nodes have the most influence on the rank among the nodes in the ANP model; determine a rank score for the nodes among the nodes in the ANP model based on an amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model; indicate the nodes determined to have the most influence; prompt a user to input refined data for the nodes determined to have the most influence while skipping input for other nodes in the ANP model; provide, for output to a user, a display showing a magnitude of rank influence for the nodes in the ANP model; base the rank of each node in the ANP model on an amount of influence each said node has on rankings of alternatives in the ANP model; and an input unit configured to input, from an input device to the processor, pairwise comparisons, ANP ratings, or ANP client data, which are stored as the data in the ANP model, the pairwise comparisons representing a judgment of priority between ANP alternatives in the pair, the ANP ratings representing a rating of a choice, and the ANP client data representing real world values, wherein the rank influence analysis is performed by finding upper and lower rank change parameter values for alternatives ordered in a weighted supermatrix representing the ANP model, wherein the amount of upward or downward change in importance that causes a change in the rank among the nodes in the ANP model is obtained by an upper rank influence score and a lower rank influence score.
 27. The apparatus of claim 26, comprising performing row perturbations in the weighted supermatrix representing the ANP model, so as to determine for the first p⁺ above fixed point p₀ where an upper rank change occurs and p⁻ below p₀ where a lower rank change occurs, and using the first p⁺ and p⁻ to construct a value that indicates the upper and lower rank influence of the rows throughout the weighted supermatrix.
 28. The apparatus of claim 26, wherein the upper rank influence score rki⁺ _(A,W,r) is defined by ${{rki}_{A,W,r}^{+} = \frac{1 - p_{A,W,r}^{+}}{1 - p_{o}}},$ and the lower rank influence score rki⁻ _(A,W,r) is defined by ${{rki}_{A,W,r}^{-} = \frac{1 - p_{A,W,r}^{-}}{p_{o}}},$ where A is the ANP model, W is the weighted supermatrix of A(of dimensions n×n), 1≦r≦n is a row of W, k is column of W, i is an alternative in A, W(p) is a family of row perturbations for row r of W with p₀ as the fixed point, p₀ represents returning the node values to the original weights, p⁺ is above p₀ where the upper rank change occurs, and p⁻ is below p₀ where the lower rank change occurs. 